FLORIDA THE STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES
ANALYSIS OF TWO PARTIAL DIFFERENTIAL EQUATION MODELS IN FLUID MECHANICS: NONLINEAR SPECTRAL EDDY-VISCOSITY MODEL OF TURBULENCE AND INFINITE-PRANDTL-NUMBER MODEL OF MANTLE CONVECTION
By YUKI SAKA
A Dissertation submitted to the Department of Mathematics in partial fulllment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Fall Semester, 2007
The members of the Committee approve the Dissertation of Yuki Saka defended on Aug. 9, 2007.
Max D. Gunzburger Professor Co-Directing Dissertation Xiaoming Wang Professor Co-Directing Dissertation
Anter El-Azab Outside Committee Member
Janet Peterson Committee Member
Xiaoqiang Wang Committee Member
The Oce of Graduate Studies has veried and approved the above named committee members.
ii
This thesis is dedicated to my mother and father.
iii
ACKNOWLEDGEMENTS
I would like to acknowledge my gratitude to those people who aected me profoundly during my life here as a graduate student. First and foremost, I would like to thank my adviser Professor Max Gunzburger. He gave me many opportunities for my development as a researcher during the course of my graduate studies. I cannot thank him enough for his generosities and support even at times when I had a diculty nding a topic. I would like to thank my co-adviser Professor Xiaoming Wang for teaching me the mathematical analysis of partial dierential equations. Without the numerous discussions I had with him about the subtlety of mathematical analysis, I would not have been able to develop my mathematical skills enough to tackle the problems in this thesis. I would also like to thank Professor Janet Peterson who has taught me the basics of the numerical analysis of PDEs. Many of my colleagues and research group members have helped me in various ways over the years. I would like to thank Dr. John Burkardt for having answers to all kinds of computer questions. I would like to thank Dr. Sang-bum Kim for interesting discussions on physics. I would like to thank all of the graduate students in my group who learned along side with me and benetted me greatly in my learning process. Lastly, I would like to thank my family for all the emotional support. My father Masakatsu has instilled in me a questioning and introspective attitude as well as the persistence to work doggedly at challenges. My mother Yumiko has given me my rst science book, and encouraged me to pursue this area. I cannot thank her enough for all of her warmth, kindness and encouragement. I thank my brother Ryuta for always being the active one. He always made our family life more interesting.
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TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. SPECTRAL EDDY-VISCOSITY MODELS 2.1 The Navier-Stokes equations . . . . . . 2.2 Eddy-viscosity models . . . . . . . . . 2.3 Spectral eddy-viscosity models . . . . . 2.4 Formal denitions . . . . . . . . . . . 3. WELL-POSEDNESS . . . . . . . . . . . . 3.1 Energy dissipation estimate . . . . . 3.2 Existence of a strong solution for N V 3.3 Further regularity . . . . . . . . . . . 3.4 Uniqueness and stability . . . . . . . 3.5 Convergence to a weak NSE solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii 1 5 5 6 8 10 12 13 16 21 22 23 25 26 27 29 32 34 37 37 41 43 43 45 48 50
4. STABILITY AND CONVERGENCE FOR SEMI-IMPLICIT SCHEME CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Existence for the nite-dimensional system . . . . . . . . . . . . . 4.2 A-priori energy estimate . . . . . . . . . . . . . . . . . . . . . . . 4.3 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Existence and convergence to NV . . . . . . . . . . . . . . . . . . 4.5 Stability and uniqueness . . . . . . . . . . . . . . . . . . . . . . .
: P=3 . . . . . . . . . . . . . . . . . . . . . . . .
5. ERROR RATE ESTIMATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nonlinear viscosity case . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hyperviscosity case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. CONTRACTION IN PHASE SPACE . . . . . . . . . . . . . . . . . . . . . . 6.1 Contraction for the N V model at p = 5 . . . . . . . . . . . . . . . . . . 2 6.2 Attractor for HV with = 3 . . . . . . . . . . . . . . . . . . . . . . . . 2 7. INTRODUCTION TO INFINITE PRANDTL-NUMBER EQUATION . . . . 7.1 One dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8. WELL-POSEDNESS OF INFINITE PRANDTL MODEL 8.1 Extension to the perodic domain . . . . . . . . . . . 8.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Maximum Principle Estimate . . . . . . . . . . . . . 8.4 Stability and Uniqueness . . . . . . . . . . . . . . . . 8.5 Regularity . . . . . . . . . . . . . . . . . . . . . . . .
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53 54 56 60 61 63 66 66 67 68 68 68 75 77 79 80 3
9. CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . 9.1 Nonlinear viscosity model . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Innite Prandtl number model . . . . . . . . . . . . . . . . . . . . . . . APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. MATHEMATICAL BACKGROUND . . . . . . . . . A.1 Multiplier theory, lter operator, and fractional gration . . . . . . . . . . . . . . . . . . . . . . . A.2 Nonlinear monotone operator . . . . . . . . . . A.3 Compactness and measure theory results . . . . A.4 Local averages and Hlder continuity . . . . . . o . . . . . . . . . . . . . . dierentiation and inte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
ABSTRACT
This thesis presents two problems in the mathematical and numerical analysis of partial dierential equations modeling uids. The rst is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the ow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the Navier-Stokes equations. In this thesis, we examine the spectral viscosity models in which only the high-frequency spectral modes are regularized. The objective is to retain the large-scale dynamics while modeling the turbulent uctuations accurately. The spectral regularization introduces a host of parameters to the model. In this thesis, we rigorously justify eective choices of parameters. The other problem is related to modeling of the mantle ow in the Earths interior. We study a model equation derived from the Boussinesq equation where the Prandtl number is taken to innity. This essentially models the ow under the assumption of a large viscosity limit. The novelty in our problem formulation is that the viscosity depends on the temperature eld, which makes the mathematical analysis non-trivial. Compared to the constant viscosity case, variable viscosity introduces a second-order nonlinearity which makes the mathematical question of well-posedness more challenging. Here, we prove this using tools from the regularity theory of parabolic partial dierential equations.
vii
CHAPTER 1 INTRODUCTION
In this thesis, we will examine two dierent problems in the analysis of partial dierential equations that arise from uid dynamics. In the rst part, we will examine some PDEs related to turbulence modeling. Fluid turbulence in three dimensions is usually modelled by the Navier-Stokes equations(NSE) with a large Reynolds number. At the present time, the simulation of this equation in this regime is a formidable task due to the need to resolve the small scale uctuations or eddies that have subtle eects on the large-scale dynamics of the uid. In order to make this problem computationally tractable, this eect must be modeled while the large-scale motion is simulated faithfully. In one idea, the velocity eld is averaged over a small radius to derive equations in terms of the averaged velocity. In this process, the problem of closure arises in that the average of the nonlinear term in NSE, which is called the Reynolds stress, must be approximated and expressed solely in terms of the averaged quantities. The way in which this is done gives rise to a variety of models. The approach we will look at, called the eddy-viscosity method, treats the Reynolds stress as a viscous eect caused by the transport and dissipation of energy due to the small-scale eddies. For this reason, this additional viscosity is called the eddy-viscosity or turbulent viscosity. The turbulence model of Smagorinsky belongs to this type [32, Smagorinsky]. For an overall survey on issues related to these models, see a discussion by [22, Layton]. Unfortunately, a simple application of this idea leads to the over-smearing of the large-scale structures in the uid. To remedy this unwanted eect, it has been proposed that the eddy-viscosity be added only to the subgrid scales. This means that we dene appropriate notion of subgrid scales, and add an eddy-viscosity only to those subgrid scales. In this way, we hope to prevent the large-scale structure from being smeared away. In this thesis, we examine a particular class of models called the spectral eddy-viscosity models in which the scales are dened in terms 1
of Fourier modes. The subgrid viscosity is simply realized as an addition of the articial viscosity only to the high-frequency modes. A simple implementation of this is to insert a high-pass spectral lter into the normal articial viscosity. We consider two types of eddy-viscosities: the hyperviscosity and the nonlinear viscosity. The hyperviscosity models are considered by various researchers because of the simplicity of the idea [25, Lions], [14, Guermond]. An example of the nonlinear viscosity is the Smagorinsky model mentioned before. The diculty of turbulence manifests itself mathematically in the unsolved conjecture of the well-posedness of 3D Navier-Stokes equations, which is one of the millennium prize problems set by the Clay mathematics institute. The hyperviscosity and typical nonlinear viscosity models are well-posed and overcome this diculty. The catch is that we have now introduced several parameters to the model: eddy-viscosity coecient, strength of the viscosity operator, and the cut-o frequency that distinguishes the small scales from the large scales. Hence, we would like to nd some guiding principles for choosing these parameters. A rigorous justication of a turbulence model is dicult partly because we do not have a good physical understanding of turbulence phenomena. However, given that the weak solution to the Navier-Stokes equations models the turbulence accurately in an ideal world where we have innite computational power, we can split the problem of turbulence modelling into two parts. One is that the turbulence model should model the uid. This can be undertaken by showing that the model is consistent with the Navier-Stokes equations in some way. For instance, this was undertaken by [14, Guermond] who analyzed and proved that the solution to the hyperviscosity model, under certain constraints on the parameters, converges to a suitable solution of the Navier-Stokes equations that satises the strongest partial regularity proved to date by [2, Caarelli, Kohn and Nirenberg]. Another type of consistency result is to assume that the solution to NSE is smooth and show that in this regime, the turbulence model is close to the solution to NSE. Intuitively, the cut-o lter plays an important role here, as it gives us a spectral convergence rate to the solution of NSE assuming that the solution is smooth. The other part is that the turbulence model should be tractable for numerical simulations. This means that it should be well-posed, and if so the numerical method used to simulate it should be stable. The by-product of these investigation is the insight gained into how various parameters introduced into the equation such as the cut-o frequency, the strength of nonlinearity or hyperviscosity, aect the consistency and 2
stability. In the second chapter we will introduce various notations and formal denitions of our turbulence models. In the third chapter, we tackle the question of well-posedness. In the fourth chapter, we consider a semi-implicit discretization of the nonlinear viscosity model. We will derive a uniform-in-time stability estimate for a bounded power input. This gives us some insight as to the reason why we should not take a strongly nonlinear viscosity. The fth chapter investigates the consistency question. We estimate the degree by which the nonlinear viscosity model is a perturbation of the NSE by estimating the error rate as the perturbation goes to zero. From this analysis, we realize that certain parameters depend on others and hence can be eliminated. In the sixth chapter, we prove that for some specic values of parameters, the hyperviscosity model and the nonlinear viscosity model have eectively nite dimensional dynamics in that any two solutions under the same forcing that agrees in the low-frequency part are exponentially contracted to a single path in the phase space; in other words, the high-frequency modes becomes irrelevant to the dynamics of the solutions. The hyperviscosity model case is shown to possess such an exponential contraction property already [37, Temam]. It is interesting that we can also estimate the dimension of such a nite dimensional attractor, in terms of the parameters. In the second part of our thesis, we consider a model related to the mantle convection in the Earths interior. A basic assumption is that in the geological time scales, the rocks will behave as a very viscous uid and hence we can use the equation of uid motion to model this phenomena. An equation that is often used to model uid convection is the Boussinesq approximation, where we neglect the density variations in the Navier-Stokes equations while we preserve its gravitational eect by modeling the buoyancy forces that are proportional to the spatial variations in the temperature eld. The temperature eld itself is convected by the uid and such a coupling provides the setting for uid convection. In order to use Boussinesq approximation to model the mantle ow, we must take the rheology of the rocks into account. In one direction, this simplies our model in that the high viscosity of the ow allows us to ignore the inertial term in the NSE: a large simplication from a mathematical point of view. Such a viscous limit of Boussinesq approximation is called the innite Prandtl number model because of the way in which the nondimensional number in the Boussinesq approximation called the Prandtl number is taken to innity in this process. This is a substantial simplication, and when the viscosity is uniform over 3
the whole domain, an extensive analysis of this model can be undertaken [3, Doering and Constantin]. Unfortunately, in our model, the viscosity variation cannot be neglected. This is because the part of the earth closer to the core is hotter and hence the rocks deform more readily than the part closer to the surface. Therefore, we model such an eect by varying the viscosity of the ow according to the the temperature eld. The innite Prandtl number model with temperature dependent viscosity is among the most popular models in use by geophysical community to simulate tectonic dynamics and mantle ows [29, Moresi]. For the constant viscosity case, the well-posedness questions are relatively easy, and large eorts are spent on the estimation of a quantity called the Nusselt number, which quanties the ratio of the heat transport due to the convection to that due to the conduction. For the temperature dependent viscosity case, the second-order nonlinearity gives an additional challenge even to the well-posedness question. In this thesis, we will show how the wellposedness can be proved for this equation. An important question that is left open is the gap that exists between the estimate of the Nusselt number between the constant viscosity case and our case.
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CHAPTER 2 SPECTRAL EDDY-VISCOSITY MODELS
In the subsequent chapters, we will introduce two spectral eddy-viscosity models and discuss various mathematical problems they inspire. investigation of isotropic turbulence. In this work we will consider a three-dimensional domain that is periodic in all directions, and thus limit ourselves to the
2.1
The Navier-Stokes equations
We will start with the famous Navier-Stokes equations.
t u u + u u = 0,
u+
= f,
where u is the velocity eld, is the pressure and = Re1 is the nondimensional inverse Reynolds number. The modern theory of partial dierential equations concerns itself extensively with dierent notions of what it means for a function to solve a PDE. The reason is that when we have stated the NSE as above, we a priori assume that the solution possess at least two spatial derivatives and one time derivative and that they are continuous. Such solutions are called classical solutions. It turns out that showing that classical solution exists for general data is a mathematically challenging task, or as in the case of NSE: unknown. If we cannot hope for nding a smooth, classical solution, can we nd a dierent notion of a solution that relaxes this criteria? The answer lies in the introduction of the weak solution that uses integration by parts so that less regularity is required of a solution. In the NSE case, a weak solution is a vector eld u such that u = 0 and satises
5
the following: t u, dt + u = 0, for all test vector eld and a test function that is smooth and compactly supported in space and in time. , denotes the duality coupling. As formulated, u is only required to have just one derivative in space. For a technical reason, the time derivative is allowed to be even more irregular in that it can be a measure. More specically, using the terminology of Sobolev spaces, u is sought in the space L ((0, T ); L2 ) L2 ((0, T ) : H 1 ) and t u, p L 3 ((0, T ) : H 1 ). Such spaces basically fall out naturally from the a priori analysis of the PDE.
4
( u :
+u
u+
) dx dt =
f dxdt
2.2
Eddy-viscosity models
We will now discuss how a typical eddy-viscosity model is derived. As noted in the previous chapter, the large-scale structure of the velocity eld can be extracted by ltering out the small uctuations. ul (x) = g u = g (x y)u(y) dy
where g is a smooth function of compact support of radius > 0. If we average the NSE using this function, we get t ul ul + + (ul ul ) l = f,
(g (u u) ul ul ) + ul = 0.
Notice how this averaging process introduces an additional stress term: R (u, u) = g (u u) ul ul . This is called the Reynolds stress. Clearly, we cannot yet solve for the average eld, because g (u u) contains interactions between the large-scale eddies and the small-scale eddies where the latter is something we would like to avoid computing. Thus, the problem of
6
turbulence modeling is that of a closure of the above model: we would like to nd an appropriate approximation S (ul , ul ) R (u, u). t w w + w = 0. The idea of the eddy-viscosity model is that the small eddies dissipate energy; therefore, their eect on the average velocity eld can be modeled by a viscous dissipation. Such an eddy-induced viscous eect is called the eddy-viscosity: S (w, w) = where T is the eddy-viscosity coecient. Smagorinsky proposed T = | u|p2 and derived the following: t u u + u u = 0. The dependence of
(w w) +
S (w, w) +
q = f,
(T w).
u
( (| u|p2 u)) +
= f,
on should be determined appropriately by dimensional analysis
by comparing with the dimensions of the Reynolds stress. We will circumvent this issue for now and simply denote this coecient as . Notice that the eddy-viscosity is modeled as a diusion eect that is proportional in strength to the velocity gradient. The degree of such a proportionality is quantied by the parameter p. The nonlinear diusion operator introduced here is called the p-Laplacian, and it is used extensively in the eld of non-Newtonian uid dynamics as well as in many other areas. The mathematical property of weak solutions to this equation is investigated by [20, Ladyzhenskaya]. It is known that the strong solution to the above equation exists globally and is unique on a periodic domain for p
11 . 5
[28,
Malek et al]. Even though this equation is well-posed in this sense, the p-Laplacian adds some diculty. In particular, we are unable to talk about the solution in a classical sense unless we know that the gradient of the velocity is continuous. The author does not know if such a result is proved. However, a local Holder regularity for an equation of the form t u (| u|p2 u) = 0 is known [8, DiBenedetto]. The proof of the local regularity for this equation depends on maximum principle type arguments. The extension of this result to the Smagorinsky case is obstructed by the fact that the maximum principle techniques 7
which are often the method of choice in proving regularity results do not apply, because of of the inherently global eect introduced by the incompressibility condition. Another turbulence model that attains well-posedness is the hyperviscosity model which was analyzed by [25, Lions]: t u u + u u = 0. The fractional dierentiation operator is dened in the frequency space as follows: () u(k) = |k|2 u(k). where u denotes the kth Fourier coecient of u. One can consider this as a certain kind of an eddy-viscosity model in which S(u, u) = | |22 u. The unique strong solution is known to exist for > 5/4 [25, Lions]. Unlike the nonlinear viscosity case, this equation has an advantage in that the strong solution is a classical solution. It is also illuminating to express this equation as an evolution equation in the frequency space. These models are derived as attempts to obtain well-posedness, a property that is lacking for the 3D NSE. Unfortunately, it is numerically observed that they tend to smear large-scale structures too much. u () u + = f,
2.3
Spectral eddy-viscosity models
In order to preserve large-scale structures, we would like to limit the regularization eect to the small-scales. In fact, this is essentially the idea in the spectral viscosity method due to [5, Tadmor] in the context of hyperbolic conservation laws, where we add a viscosity to only the high-frequency part. Inspired by this, we may propose the following spectral viscosity model for NSE: t u u + u u = 0. 8 u Q( (Q( u)) + = f,
where Q is a high-pass lter ; that is, it erases all the low-frequency modes of the input. Therefore, the subgrid scale viscosity is eective only for the high frequency part of the solution. An important point is that for a smooth solution that can be expressed in terms of low modes, the spectral-viscosity operator becomes zero. Thus, low-mode solutions in fact satisfy NSE. This indicates how the ltered viscosity tries to model the large-scale structure in the uid consistently. In the above, the turbulent viscosity was constant, but this will not make the model well-posed. For this reason, we will combine this spectral idea with eddy-viscosity. Basically, we modify the Smagorinsky model and the hyperviscosity model so that the regularization only aects the high-frequency modes: t u u + u u Q (| Qu|p2 Qu) + = f, (2.1)
t u u + u
u () Qu +
= f.
(2.2)
As it turns out, it is benecial to stabilize 2.1 by a linear ltered viscosity as well: t u u + u u Q( ((1 + |Q( u)|p2 )Q( u)) + = f. (2.3)
Such an idea is used for the numerical simulation of turbulence in [16, Jansen] in the setting of two-grid nite element method and where the cut-o operator is dened by an appropriate orthogonal projection onto the ne scales. Notice that the spectral viscosity model inevitably introduces several free parameters. These parameters are important in quantifying the trade-os in modeling turbulence. The large nonlinearity or hyperviscosity and low cut-o gives the model more stability at the expense of increasing the modeling error, while small nonlinearity or hyperviscosity and high cut-o increases the consistency for less stability. The purpose of our work is to analyze 2.3 and 2.2 and mathematically investigate such a trade-o. It is also worth noting that the model 2.2 is natural in our periodic setting because the equation can be formulated in the frequency space. This makes the analysis of 2.2 somewhat simpler than 2.3, where it is dicult to interpret the spectral nonlinear viscosity either in the frequency space or the physical space.
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2.4
Formal denitions
We now formalize the models 2.3 and 2.2. Let I = [0, T ] and QT = I T3 denote the time-space cylinder in a periodic domain. T3 is the unit box [0, 1]3 with identication of the planes xi = 0 with xi = 1. To formalize the various aspects of this, we dene the projection operator: PN by PN (f ) =
|k| N
f (k)eikx .
Dene P to be the Leray projector, an orthogonal projector onto the space of divergence free vector elds. Dene XN = PN (L2 (T3 )), VN = P PN ((L2 (T3 ))3 ). We dene the lter QM as I PM . Thus, the nonlinear viscosity model is the following: t u u + u u = 0. u QM ( (|QM ( u)|p2 QM ( u)) + = f, (2.4)
Clearly, this model is determined by three parameters M , p and . Thus we call the above model N V ( , p, M ). The hyperviscosity model is the following:
t u u + u u = 0.
u () QM u +
= f, (2.5)
Clearly, this model is determined by three parameters M , and . The operator () is dened as a multiplier in the Fourier space: () = |k|2 , where k is the wave-number. Note that this is basically a generalization of dierential operators to the setting with fractional index. One caveat is that such operators, unlike the Laplacian, are global (in the physical space) for fractional index, hence presenting some additional subtlety in the analysis. We call the above model HV ( , , M ). For a notational convenience, we set u = QM u, u = PM u and refer to the former as the high-frequency part of u and the latter as the low-frequency part of u. Now, even though we can show that a classical solution exists for 2.5, this is not necessarily so for 2.4. Therefore, 2.4 does not make sense as stated. Hence, we must introduce the notion of weak solutions to N V and HV . 10
We call u a weak solution to N V ( , p, M ) if (C (QT ))n , = 0 and
u = 0 almost everywhere and for all
= 0, and for almost all time, +u u + | u|p2 u : dx = f dx, = 0 and
t u + u : and HV ( , , M ) if
u = 0 almost everywhere and for all (C (QT ))n ,
= 0, and for almost all time, t u + u : where | | is dened as: | u(k) = |k| u(k). | Our objective is to solve these equations by approximating them by a nite-dimensional system that solves a similar problem. In other words, call uN a Galerkin N -dimensional projection of u and call it the solution to the system N VN ( , p, M ), if u L2 ((0, T ) : VN ), t u Ls ((0, T ) : VN ) for some s > 1 and for all L2 ((0, T ) : VN ) and almost all time, +u u + | | u | | dx = f , dx,
t uN + uN :
+ uN
uN + | u |p2 u : N N
dx =
f dx,
and HVN ( , , M ) if u VN and for all L2 ((0, T ) : VN ) and almost all time, t uN + uN : + uN uN + | | u | | dx = N f dx.
Now that all the formal denitions are in place, we are in good shape to tackle, in the next chapter, the important mathematical question of well-posedness of our model.
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CHAPTER 3 WELL-POSEDNESS
In this chapter we will discuss well-posedness issues for the spectral viscosity models. In general, the well-posedness of a given partial dierential equation means that it possesses the following properties: 1. A solution to the equation exists in an appropriate (weak) sense. 2. This solution is unique. 3. The solution is regular in an appropriate sense. The point of introducing a weak solution is that the weaker the class in which we seek a solution, the easier it is to nd it by the compactness method. In certain cases such as hyperbolic conservation laws, the introduction of a weak solution is inevitable, because we know that there exist non-classical solutions. For the Navier-Stokes equations, we know that a weak solution exists, but we are neither sure of its uniqueness nor its regularity. For a detailed discussion of the well-posedness for the Navier-Stokes equations, we refer the reader to [36, Temam] and [4, Constantin and Foias]. The well-posedness is also important from a physical standpoint. The Navier-Stokes equations are derived from a microscopic model by statistical averaging; therefore, we assume that such a procedure is valid. However, if the macroscopic model that results possesses a weak solution with a singularity, then it may undermine the very process of averaging that took us from the microscopic to the macroscopic. This is one of the reasons why the regularity results for the equations of mathematical physics hold particular interest for mathematicians. Proving the well-posedness of our model is also an important undertaking from a numerical simulation viewpoint. This is because of the various stability estimates that the 12
well-posedness proof generates as a by-product. These stability estimates oer a skeleton for proving that the numerical scheme that is generated from the turbulence model is wellconditioned. Now the classic result in the direction of showing well-posedness for NSE is the result of J. Leray on the existence of a weak solution to the Navier-Stokes problem. Theorem 3.0.1 Let f L2 (QT ), u0 H 1 and
4 3
u0 = 0, then there exists a weak solution u = 0 almost everywhere and
u L2 (I; H 1 ) L (I; L2 ) and t u L (I; H 1 ), such that for all C (QT ) with = 0 and
Tn
dx = 0, u dxdt =
QT
t u + u :
QT
+u
f dxdt.
We currently do not know whether a weak solution is unique. The question of uniqueness is intimately tied to the regularity question as there are a host of results that show that if a solution possesses a certain amount of regularity then it is unique. Examples of results in this direction are that of [30, Prodi] and [31, Serrin].
3.1
Energy dissipation estimate
To discuss the well-posedness issue in more detail, we note that one of the diculties associated with the Navier-Stokes equations is its lack of many globally controlled quantities. What the turbulence model does is that it basically adds another globally controlled quantity to the Navier-Stokes equations so that the global well-posedness is restored. Suppose N V and HV possess a smooth solution, then if we test N V and HV by u, we get the most important global dierential inequality called the energy dissipation estimate: 1 t u 2 and 1 t u 2 Let C0,f = u(0) +
0 2 2
+
u
2
+ (
u
2
+
u p ) = (f, u), p
2
+
u
2
+
| | u
t
= (f, u).
f ds.
This implies the following energy balance for N V ( , p, M )
t t
u(t)
2
+
0
u
2
+
0
( 13
u
2
+
2 u p ) (3/2)C0,f , p
(3.1)
and for HV ( , , M )
t t
u(t)
2
+
0
u
2
+
0
| | u
2
2 (3/2)C0,f .
(3.2)
Notice how the additional control on norms of u is added in addition to the usual energy balance equation for the NSE. Another important bound for both models as well as for NSE is that for uniformly bounded forcing, the L2 norm of the solution remains bounded. This can be seen from the inequality: t u + u f , which follows from the energy dissipation estimate and application of the Poincar inequality e to the viscosity part. Consequently,
t
u(t) et u(0) +
0
e(ts) f (s) ds et u(0) +
f
L ((0,t):L2
.
3.1.1
Why is it dicult to show well-posedness for NSE?
Terence Tao [34] indicates why the well-posedness problem for NSE is dicult by giving useful dimensional heuristics for intuitively grasping these types of global energy estimates. Suppose we take the forcing f = 0 and that u is supported in the frequency support of order N and hence has a physical support of order N 1 by the uncertainty principle. Thus, the energy dissipation estimate tells us that the following global quantities are controlled:
Td
|u|2 U 2 N d and
T 0
Td
| u|2 T N 2 U 2 N d where we have designated their dimension.
T 0 d
Due to the skew-symmetry of the nonlinear term, its eect on the globally conserved quantities is nil; however, it can have a local eect of order Now, let us assume that u(0) O(1), then u(t) follows that the dissipative eect is of order is of order
T 0 T 0 Td 2
u
u u U 3 N N d .
U 2N
2
O(1) so U N d/2 . It
u
T N 2 , while the nonlinear eect
u
u u T N 1+d/2 .
When d = 2, the linear term and the nonlinear term have the same dimension. Hence, we say that our energy dissipation bound is critical for the 2D NSE, while for d = 3, the nonlinearity dominates and hence we call our energy bound supercritical. The problem with 3-dimensional turbulence is that the global energy bound is supercritical and does not provide enough control on the size of u to prevent the local instability due to the nonlinearity from happening. 14
Notice that energy dissipation also says that T N 2 O(1) or T N 2 , which is the wellknown parabolic scaling T L2 . This roughly means that the solution can have the N th mode staying large only for time interval of length N 2 . This implies that the nonlinearity is of the order T N 1+d/2 N d/21 . This is a positive power of N for d 3. Hence, it leaves the possibility for energy to be cascaded to higher and higher frequencies only staying within any one frequencies in time of interval length N 2 , and meanwhile increasing the nonlinearity. But sum of N 2 as N is nite; therefore, the solution can blow up in nite time due to ever increasing nonlinear destabilizing eect. However, no such blowup is known to happen and hence this problem remains open. On the other hand, our turbulence models add more dissipative terms. For N V model we have
T 0 Td
| u|p T U p N p N d T N d( 2 1)+p which for 3D becomes critical at p =
T 0
p
11 . 5
Upon actual analysis, this equation is shown to possess a strong solution at this value of p. For the HV model, we have
Td
|| | u|2 T U 2 N 2 N d T N 2 which becomes critical
at = 5 , an index beyond which the well-posedness is established. 4 Notice how the simple dimensional analysis is a powerful tool in predicting the criticality of the equation, and therefore gives us forsight into the well-posedness question. In any case, from the dimensional analysis, our turbulence model can be thought of as adding a control on the critical quantities for the high-frequency modes of the equation so that the possibility of a blow-up due to cascading is prohibited. However, the nonlinearity is free to act on the low-frequency part due to ltering, and hence we expect that the low-frequency behavior models the uid well.
3.1.2
Simple interpolation result
L ((0,T ):L2 )
Note that the energy balance equation roughly says that u
and 1/2
u
L2 ((0,T ):L2 )
are of the same order. This implies that we should have a bound on the spaces that are in between these spaces. This is the content of the following interpolation lemma. Lemma 3.1.1 If u is a smooth solution to N V , or HV , and 2 r 6 then u
< 4r L 3(r2) ((0,T ):Lr ) 3(r2) 4r
u
6r 2r L ((0,T ):L2 )
u
3(r2) 2r L2 ((0,T ):L2 )
C0,f
15
Proof Let s =
4r , 3r6
then
t
u
0
s r
dt
u
s 6r 2r 2
u
s 6
3(r2) 2r
dt
u
2(6r) 3(r2) L ((0,T ):L2 )
u
2 L2 ((0,T ):L2 ) ,
where we have used the Hlders inequality and the Sobolev inequality. o
3.2
Existence of a strong solution for N V
11 , 5
Having heuristically discussed the well-posedness issues, we will see in this section that precisely beyond the index p as we saw from the dimensional analysis, N V possesses a strong solution. We will also show that the weak solution to N V converges to a weak solution of NSE as 0. For weak solutions, regularity and convergence to NSE of the HV model, we refer the reader to [14, Guermond]. To this end, we will proceed in a standard fashion by deriving a series of a-priori estimates. Besides the energy dissipation estimate 3.1, we must show regularity in time and in space. First, we will show regularity in time: Lemma 3.2.1 Let
1 q 1 + p = 1. If uN is a solution to N VN then t uN Lmin{4/3,q} (I; W 1,q ).
Proof If VN , then, (t uN , ) = (f + uN uN = (f + uN uN + Q M (| QM uN |p2 QM uN ), ) (| QM uN |p2 QM uN ), )
(uN uN ) + QM
= (f, ) + ( uN + uN uN , ) ( | QM uN |p2 QM uN , QM )
< <
( f ( f
1 1
+ +
uN + uN uN + uN
2 4) 2 4)
2 2
+ +
Q M uN Q M uN
p1 p p1 p
QM
p
QM p ,
where we have used the (p, p)-estimate A.1.6 for the operator QM = I PM . After time integration, (t uN , ) dt ( f + uN
1
+
uN )
L (I;L2 )
2 L8/3 (I;L4 )
L4 (I;L2 )
+
Q M uN
p1 Lp (I;Lp )
Lp (I;Lp ) .
Therefore, the above inequality tells us that t uN Lmin{4/3,q} (I; W 1,q ). 16
Now, in order to prove the existence, we will show that the Galerkin projection satises the following spatial regularity result uniformly in N . The proof of this theorem is almost identical to the special case of that given in [28, Malek et al] except that in our case we have a cut-o lter that needs to be handled in a special way. Theorem 3.2.2 Let uN be the solution to N VN ( , p, M ) where p Let =
2(3p) . 3p5
11 . 5
Then if p < 3,
t 2
<
uN (t) while if p 3,
uN (0)
2
+
0
(
f
2
t
+ M2
uN
p p)
+(
0
2 3
uN
p 1/(1) , p)
t
uN (t)
2
<
uN (0)
2
+
0
f
2
+
uN
3 3
dt.
We also have that uN Lp (I; W 1,3p ) L2 (I; H 2 ) L (I; H 1 ). Proof We multiply the equation by uN : 1d 2 dt uN
2
+
(
(1 + | u |p2 ) u , u ) N N N
2
+ uN uN 4 Note that uk k uj (ll uj ) = Consequently, l uk k uj l uj +
+ b(uN , uN , uN ) 1 2 + f 2.
1 2
uk k (l uj )2 =
l uk k uj l uj .
b(uN , uN , uN ) Now, let Ip (u ) = N A.2.3. (
i,j,k
uN
3 3.
| u |p2 (kj u i )2 dx. Then due to the monotonicity formula N N | u |p2 u , u ) Ip (u ) N N N N
thus, we have our inequality of the form 1d 2 dt uN
2 2
+ (Ip (u ) + u N N 17
2
) + uN
2
<
uN
3 3
The p 3 case basically follows from the above equation. We must work a little bit harder for p < 3. Note that by interpolation, uN and uN Thus given 0 < < 1, uN
3 3 3 3
uN
2(p1) 3p2
2
uN
p 3p2
3p
,
uN
p
p1 2
uN
3p 2
3p
.
uN
Q1 2
uN
Q2 p
uN
Q3 3p ,
p where Q1 = 3 (p1)2 , Q2 = 3(1 ) p1 , Q3 = 3(1 ) 3p + 3 3p2 . Recall A.2.4: 3p2 2 2
uN Using these inequalities, we obtain 1d 2 dt uN
2
3p
<
I(uN )1/p .
+
<
(Ip (u ) + u N N 1 1 1 1 f f f f
2
2
) + uN uN uN uN
Q1 2 Q2 p Q2 p (
2
+ + + +(
uN uN uN
Q3 p
Q1 2 Q1 2 Q1 2
uN u N
Q3 3p 3p
2
+Mp
2
2
uN uN
Q3 p) Q3 p)
1 2
<
2
1 Q2 p p (Ip (uN ) p
+Mp
<
2
uN
uN
Q2 pQ3 p )
+ (Ip (u ) p + M p N 2
uN
p p) ,
where we have used the Bernstein inequality A.1.7 on the third line and the previous lemma on the fourth line. It is now clear which must be chosen. We would like to set Q2 We get that 1= and = (3 p)(3p 2) . 6(p 1) 18 p(3p 5) , 6(p 1) p = p. p Q3
Consequently, Q1 = 3 p, Q2 = We have
p(3p5) , 4
Q3 =
(3p)3p . 4
p 4 p = = . p Q3 Q2 3p 5 Now set = Then note that Q3 p 3(3 p) 2 = = . p p Q3 3p 5 3 Thus, we get 1d 2 dt uN
2
Q1 p 2(3 p) = . 2 p Q3 3p 5
+
<
Ip (QM uN ) + 1 f
2
QM uN ( uN
2
+ uN uN
p p
2
+C
2 3
2 2)
+ (Ip (u ) + M 2 N 2
uN
p p)
Therefore integrating in time, uN (t) 2 uN (0) 2 t 2 1 f 2 + 3 ( uN + C 0 Let A =
t 1 0
2 2)
uN
2 3
p p
+
1 M2 2
2 2)
uN uN
p p
dt. If g = uN
2
f
2
+
1 2
M2
uN
p p
dt, and B =
(
uN
p p.
,
the above inequality is of the form
t
g(t) g(0) A +
0
Bg dt.
This inequality is solved to give the following:
t t
f (t) ((f (0) + A)1 + (1 )
0
B dt)1/(1)
<
f (0) + A + (
0
B dt)1/(1) .
Applying this to above we get,
t
uN (t)
2
<
uN (0)
2
+
0
(
f
2
t
+ M2
uN
p p)
+(
0
2 3
uN
p 1/(1) . p) 11 . 5
This implies that
uN L (I; L2 ) given that 1 which translates to p
This would then imply the further regularity stated in the theorem.
19
From the energy dissipation estimate and the regularity of the time-derivative, we have that uN L (I; L2 ) L2 (I; H 1 ), t uN Lmin{4/3,q} (I; W 1,q ) . We now use Aubin-Lions compactness theorem to obtain an appropriate subsequence which we still call uN such that uN u weakly in L2 (I; H 1 ), uN u strongly in L2 (I; L2 ). The regularity result shows that in fact uN Lp (I; W 1,3p ) L (I; H 1 ) L2 (I; H 2 ) uniformly in N , and therefore the same holds for u. In particular, Aubin-Lions theorem implies the strong convergence in L2 (I; W 1,q ) for q < min{6, 3p}. Now let j Xj . Then, clearly by the weak convergence and incompressibility condition, (t uN , j ) t u, j , and ( uN , j ) Finally, since uN u, j .
u in L2 (I; L2 ), and uN u strongly in L2 (I; L2 ) we have, (uN uN , j ) u u, j .
The convergence of the p-Laplacian part is slightly more challenging. If p < 6 then, note that due to A.2.2, | u |p2 u | u|p2 u, N N j (p 2) j
p
(u u) p ( N
u
p
+
u N
p2 . p)
Due to the fact that p < 6, uN u in L2 (I; W 1,p ) strongly. The Jackson inequality A.1.9 tells us that QM is an (p, p) type operator. Therefore, u u in L2 (I; W 1,p ). Thus, we N have the required convergence. For p 6 we must consider a dierent approach. First, since due to A.3.1 there exists a subsequence Ni such that Note then that for any set M QT , | u i |p2 u i : N N
M
u N
u in L2 (QT ),
u i N
u almost everywhere.
Consequently, due to A.2.2, | u i |p2 u i | u|p2 u almost everywhere. N N
j |M |1/p
u i N
p1 p
j
.
Thus, the p-Laplacian term is uniformly integrable. Thus, the required convergence follows from A.3.2.
20
Theorem 3.2.3 If 11 . 5 Then there exists a weak solution u to N V ( , p, M ) given that u(0) H 1 and f L2 . In p fact u possesses a further regularity: u Lp (I; W 1,3p ) L2 (I; H 2 ) L (I; H 1 ).
3.3
Further regularity
We will show in this section that t u L2 (I; L2 ) and u L (I; Lp ), given that u(0) W 1,p . Thus with this additional result, u satises a host of regularity results and we may call such a solution the strong solution to N V . Theorem 3.3.1 With the same hypothesis as in the previous section and suppose in addition that u(0) W 1,p . Then, the solution possesses a further regularity that t u L2 (I; L2 ) and u L (I; Lp ). Proof We multiply the equation by t u to obtain, t u
2
+ +
((1 + | u|p2 ) u, t u)
t u 2 + b(u, u, t u) dt 2 1 t u 2 + 4 f 2 dt. 4 Using A.2.2 we have 1 2 t u
2
+
<
( ( +
u(t) u(0)
2 2
+ +
u(t) p ) + p u(0) p ) + p u|2 dx + 4 f
2
u(t) u(0) dt.
2 2
|u
We have that |u u|2 dx u
2
6p 3p2
u
2 3p
<
u
2 2
u
2 3p . 6p 3p2
The last inequality is due to the Sobolev inequality and the fact that Thus, we have 1 2 t u
2
6.
+
( ( +
u(t) u(0) u
2 2
+ +
u(t) p ) + p u(0) p ) + p u
2 L2 (I;L3p )
u(t) u(0)
2 2 2
2 L (I;L2 )
+4
f
dt.
21
The right hand side is bounded due to the regularity result of the previous section. Therefore, the theorem follows.
3.4
Uniqueness and stability
In this section we show that the solution to N V ( , p, M ) is unique. We will do this by deriving a stability estimate. Such an estimate also holds for the 3D NSE given that a solution is suciently regular. The key point is that due to the regularity result proved in the previous section, the uniqueness for the N V follows. Theorem 3.4.1 Let u1 and u2 be two distinct solutions to N V ( , p, M ), such that p Then, (u1 u2 )(t)
2
11 . 5
u1 (0) u2 (0) 2 e
Rt
0
min{ 3
u1
4,
M 2+
3
u1
4 } ds
Since u1 L (I; H 1 ) for the range of index satised by p, the solution to N V ( , p, M ) is unique. Proof We would like to derive an estimate for w = u1 u2 . Since u1 and u2 satisfy t ui ui P QM we can see that w satises t w w P QM +P (u1 u1 u2 ((1 + | u1 |p2 ) u1 ((1 + | u2 |p2 ) u2 ) u2 ) = 0. ((1 + | ui |p2 ) ui ) + P (ui ui ) = fi ,
We test this equation against w and use the monotonicity A.2.2 to get, 1 t w 2 + w 2 (w u1 , w).
2
+ (
w
2
+
w p) p
First note that due to Gagliardo-Nirenberg, and w
4
w = 0 we have
3/4
w
1/4
w
.
22
Thus, (w 12 12 u1 , w) w w
2 2
u1
3
w u1
4
3/2
w
2
1/2
+
u1
w
2 4
+C
w
+ M (1/21/4)32 u1 4 ) w 2 ,
u1
w
2
+ (M 3/2
u1 +
3
where we have used the Bernstein inequalityA.1.7. We also have that (w u1 , w) u1 w 3/2 w w 2 + 3 u1 4 w 2 . 12 Consequently, t w C
2 1/2
12
w
2
+ C 3
u1
4
w
2
+ u1
p p
w
2
+ (
w
2
+
w p) p
3
+ C2 (min{ 3
u1 4 , M 2 +
u1 4 }) w 2 .
Therefore, the Gronwall inequality implies t (e
Rt
0 (min{ 3
u1
4,
M 2+
3
u1
4 }) ds
w 2) +
w
2
+ (
w
2
+
w p ) 0. p
Using forward inequality, this implies the following estimate: (u1 u2 )(t)
2 Rt
0
u1 (0) u2 (0) 2 e
min{ 3
u1
4,
M 2+
3
u1
4 } ds
.
3.5
Convergence to a weak NSE solution
We have shown that the N V model is well-posed. This indicates at least in part that solving the N V equation numerically is tractable. We must however also show that somehow N V models the turbulence well. For this, we show that the solution to N V converges to the solution of NSE. The rst issue we must clarify, however, is that there are two senses in which the sequence of solutions to N V can be possibly taken to converge to a NSE solution. One is to take M and the other 0. In fact, we are also interested in cases in which both of these occur at the same time. In light of the fact that we do not know much about 23
the regularity of a weak solution to NSE, it is unlikely that as M independent of , the p-Laplacian term goes to zero. Therefore, the most sensible thing to do is to take the sequence as 0, and perhaps let M as a function of . Thus, we will show that the sequence u parametrized by NSE, and let M depend on contains a subsequence that converges to a weak solution of by expressing it as M ( ). and
Note that u L2 (I; H 1 ) L (I; L2 ) and t u Lmin{4/3,q} (I; W 1,q ) uniformly in such a u is a weak solution of NSE. Note that, ( | QM ( ) u|p2 QM ( ) u, We know that
1/p
therefore we can use the Aubin-Lions to obtain the weak limit u. We want to show that
)
1/p
(
1/p
QM ( ) u p )p1
p.
QM ( ) u
p
is uniformly bounded in
due to 3.1. Thus, we may
take 0 and see that the nonlinear term goes to zero. Therefore the limit of the solution to N V ( , p, M ( )) as 0 satises the weak formulation for the NSE. Lemma 3.5.1 Let u be the solution to N V ( , p, M ( )). Then there exists u such that a subsequence of u converges weakly u j
j
u in L2 (I; H 1 ) and u j u strongly in L2 (L2 ) as
0, and u is a weak solution of the NSE. In this chapter we proved some preliminary results concerning the well-posedness. In the
next chapter, we will investigate the N V model further by discretizing the equation in time. We will derive a certain stability result that allows us to address the question of choosing the appropriate parameter for the model.
24
CHAPTER 4 STABILITY AND CONVERGENCE FOR SEMI-IMPLICIT SCHEME : P=3 CASE
In this section, we consider a semi-implicit scheme to solve the nonlinear spectral viscosity method for p = 3. We call this method semi-implicit in the sense that we only lag the nonlinear inertial term in time, but keep the p-Laplacian implicit. First, assume that f L2 ((0, T ) : L2 ) L1 ((0, T ) : L2 ). Suppose n is the number of mesh points. Let t > 0 be given, and ti = ti. Given u0 W 1,3 , seek ut (i) : {i = 1, . . . n} T3 R, i > 0 and ut (i) = 0, u(0) = 0 such that, ut (i 1) dx ) + ut (i) : + ut (i 1) ut (i) dx (4.1)
ut (i) dx +t t
((1 + | ut (i)|) ut (i) : f (it) dx = 0.
We call the system of such equations for i = 1, . . . n, N Vt ( , 3, M ). Notice that we are lagging the nonlinear convection term, so that we are essentially solving a nonlinear Oseen type equation. The goal of this section is to prove the stability and convergence of the semi-implicit scheme in this setting. p = 3 turns out to be especially nice since for the equation that is satised by ut (i), the L3 norm of the gradient has exactly the same dimension as the nonlinear form. We will use this property to derive estimates that allows us to show convergence as t 0 to the solution of N V and show the rate at which the convergence takes place. We will note that the semi-implicit Euler scheme for the case M = 0 has been analyzed by [7, Diening]. However, their emphasis was on a more physically signicant(and challenging) 25
case of smaller ps. We rst proceed to discretize the system and show existence to the nite dimensional approximation.
4.1
Existence for the nite-dimensional system
We will rst show that the semi-implicit equation is solvable. To this end we will use the Galerkin projection of ut (i) much as was done in the previous chapters. Let fi,N = PN f (i). In fact we want to solve for ut,N (i) VN such that for each VN . ut,N (i) dx t t ut,N (i 1) dx + ) + ut,N (i) : + ut,N (i 1) ut,N (i) dx (4.2)
((1 + | ut,N (i)|) ut,N (i) : fi,N dx = 0.
Let us call the above equation N Vt,N ( , p, M ). Alternatively, we can set fi (ut,N (i))k with fi : RN RN to be the left-hand-side of the above equation for each of the basis k VN . So that the above equation can be stated as an algebraic equation: Find ut,N (i) so that, fi (ut,N (i)) = 0. Now, if we test this equation against ut,N (i), we get fi (ut,N (i)) ut,N (i) = ut,N (i)
2
+t( ( ut,N (i) 2 + ut,N (i) 3 ) + ut,N (i) 2 ) (ut,N (i 1), ut,N (i)) 3 1 ut,N (i) 3 ) + ut,N (i) 2 ) ut,N (i) 2 + t( ( ut,N (i) 2 + 3 2 1 ut,N (i 1) 2 + t fN ut,N (i) . 2 Thus, 1 fi (ut,N (i)) ut,N (i) ( ut,N (i) 2
2
ut,N (i 1) 2 ) t fN
ut,N (i) .
We note then the following topological xed-point theorem taken from [9, Evans] which is a consequence of the Brouwers xed point theorem. It says roughly that any continuous 26
vector eld that points outward on each point of a suciently large sphere must equal zero somewhere inside that sphere. Another name for this type of theorem is a Hairy-ball theorem.
Lemma 4.1.1 Assume the continuous function f : RN RN satises f(v) v 0, if |v| = r for some r > 0. Then there exists a point v B(0, r) such that f(v) = 0. Using such an assertion, together with an assumption that t is suciently small, we can prove the existence of ut,N (i) for each i = 1 . . . n such that 1 ut,N (i) 2 + t( ( ut,N (i) 2 + ut,N (i) 2 1 ut,N (i 1) 2 + t fN ut,N (i) . 2
3 3
+
ut,N (i) 2 ) (4.3)
4.2
A-priori energy estimate
Now, having shown the existence for the nite-dimensional system we now want to see what happens as N . For this, various a-priori estimates must be shown. The energy dissipation estimate in this setting appears as a discrete iteration formula; and therefore, we must introduce the following discrete Gronwall type inequality to solve it: Lemma 4.2.1 Suppose we have a sequence ci > 0, ai > 0, bi > 0 that satises ci + ai i ci1 + bi , for each i = 1 . . . n. Then for each j = 1 . . . n,
j j j j j
cj +
(
i=1 l=i+1
l )ai (
l=1
l )c0 +
(
i=1 l=i+1
l )bi .
Proof We will proceed by induction. The base i = 1 case follows from the hypothesis. Suppose that the assertion is true for j = 1 . . . n 1, we would like to prove it for j = n. We
27
have
n n n1 n1
cn +
(
i=1 l=i+1
l )ai = cn + an + n
n1 n1
(
i=1 l=i+1
l )ai
n1 n1
n (cn1 +
n
(
i=1 l=i+1 n
l )ai ) + bn n (n1 c0 +
n1
(
i=1 l=i+1
l )bi ) + bn
=(
l=1
l )c0 +
(
i=1 l=i+1
l )bi .
We get as a result the existence of a solution to 4.2 that satises the discrete energy dissipation estimate. Lemma 4.2.2 Let C0,f = u0,N + (
i=1
j
fN (i) t).
Then, there exists a solution ut,N (i) to 4.2 which satises,
j
max ut,N (k)
kj
<
2
+
i=1
2t( (
ut,N (i)
2
+
ut,N (i)
3 3
+
ut,N (i) 2 )
2 C0,f .
Proof First sum up the 4.3 for all i = 1 . . . j to obtain
j
ut,N (j) uN (0)
2
+
i=1
2t( (
j
ut,N (i)
2
+
ut,N (i)
3 3
+
ut,N (i) 2 )
2
+2
i=1
t fN (i) ut,N (i) .
Taking the maximum over j on the left hand side, and canceling, we get 1 max ut,N (j) j 2 uN (0)
2 j 2
+
i=1 j
2t( ( fN (i) t)2 .
ut,N (i)
2
+
ut,N (i)
3 3
+
ut,N (i) 2 )
+ 4(
i=1
28
4.3
Regularity
In order to show existence, we would like to use a compactness method. To do this we must show that the solution has an additional regularity.
4.3.1
Space-regularity
Csp =
1 2 2 3 C0,f + C0,f M 2 1 + 4C0,f ,
5
Theorem 4.3.1 Let
then a solution to 4.2 satises, 1 max ut,N (i) 2 + t( ( ut,N (i) 2 + ut,N (i) p ) + ut,N (i) 2 ) 3p jn 4 1 < uN (0) 2 + Csp + O(t). 2 Proof We multiply 4.2 by ut,N (i). 1 ut,N (i) 2 + t( ( (1 + | ut,N (i)|) ut,N (i), ut,N (i)) 2 + ut,N (i) 2 + t(ut,N (i 1) ut,N (i), ut,N (i))) 1 ut,N (i 1) 2 + t fi,N ut,N (i) . 2 Note then ut,N (i 1)
<
ut,N (i) (ut,N (i)) =
l,k,j
l ut,N (i 1)k k ut,N (i)j l ut,N (i)j .
ut,N (i)
3 3
+
ut,N (i 1)
3 3
We obtain our inequality of the form 1 ut,N (i) 2 + t( ( ut,N (i) 2 + I3 (ut,N (i))) + ut,N (i) 2 ) 2 2 1 < ut,N (i 1) 2 + t( ut,N (i) 3 + ut,N (i 1) 3 ) + t fi,N 3 3 2 Now we have u Therefore, 1 2
3 3
<
ut,N (i) .
M2
3
u
3 2
<
M2
5
u
2
u
ut,N (i)
2 2
+ t( ( ut,N (i)
2 2
2
+ I3 (ut,N (i))) + ut,N (i) 2 )
3 3
1 ut,N (i 1) 2 5 +tM 2 ( ut,N (i)
+ t( +
ut,N (i) ut,N
2
+
j
ut,N (i 1) 3 ) 3 ut,N (i) .
(i 1)) max ut,N (j) + t fi,N 29
We now sum over i to obtain, 1 2
<
ut,N (i)
2
+
t( ( ut,N (i) ut,N M2
5
2
+ I3 (ut,N (i))) + ut,N (i) 2 )
1 uN (0) 2 + 2 + max ut,N (j)
j
3 3 t 2
ut,N
+(
fi,N t) max
j
ut,N (j) + O(t)
Consequently, using 4.2.2, max
jn
<
1 4
ut,N (i) uN (0)
2
2
+
1
t( ( ut,N (i)
5
2
+ ut,N (i)
p 3p )
+ ut,N (i) 2 )
1 2
+
2 3 2 C0,f + C0,f M 2 1 + 4C0,f + O(t)
4.3.2
Regularity in time
To use the compactness method, we also need that ut,N has some regularity in time. The following lemma provides such a result. Lemma 4.3.2 Let Ctm = then,
n 1 6 2 Csp + C0,f 1 + M 3 C0,f 1 + 2
fi,N
2
t.
(t)1 ut,N (i) ut,N (i 1)
i
2
+( (
1 2 1 < ( ( 2
ut,N (j)
1 3 1 ut,N (0) 2 + 3
2
+
ut,N (j) 3 ) + 3
4 ut,N (0) 3 ) + 3 4
ut,N (j) 2 ) ut,N (0) 2 ) + Ctm . + O(t)
Proof We multiply 4.2 by
ut,N (i)ut,N (i1) . t 2
(t)1 ut,N (i) ut,N (i 1)
+ ((1 + | ut,N (i)|) ut,N (i), ut,N (i) ut,N (i 1))
+( ut (i), (ut,N (i) ut,N (i 1))) +t(ut,N (i 1) ut,N (i), (t)1 (ut,N (i) ut,N (i 1)))
t(f, (t)1 (ut,N (i) ut,N (i 1))) = 0. 30
Consequently,noting that | u| u : v 2 3 u
3 3
+
1 3
v 3, 3
1 1 1 (t)1 ut,N (i) ut,N (i 1) 2 + ( ( ut,N (i) 2 + ut,N (i) 3 ) + 3 2 2 3 2 1 1 ( ( ut,N (i 1) 2 + ut,N (i 1) 3 ) + ut,N (i 1) 2 ) 3 2 3 2 +t( ut,N (i)ut,N (i 1) 2 + ut,N (i)ut,N (i 1) 2 + fi 2 ) Note that by Hlder inequality, o | u|2 |u|2 dx u but we have by Gagliardo-Nirenberg, u We also have, | u|2 |u|2 dx u
2 2 6
18 7
ut,N (i) 2 )
2
18 7
u
2 9
<
u
6
18 7
+
u
3 9
u
4
u 2.
u
2
u
2
u 2M 3
where we have used the Sobolev inequality and the Bernsteins A.1.7 inequality. Therefore, 1 1 1 (t)1 ut,N (i) ut,N (i 1) 2 + ( ( ut,N (i) 2 + ut,N (i) 3 ) + 3 2 2 3 2 1 1 2 3 2 < ( ( ut,N (i 1) + ut,N (i 1) 3 ) + ut,N (i 1) ) 2 3 2 +t( ut,N (i) 3 + ut,N (i 1) 2 ut,N (i 1) 4 9 +M 3 ut,N (i 1)
2
ut,N (i) 2 )
ut,N (i)
2
+ fi 2 )
If we sum this successively, we obtain
n
(t)1 ut,N (i) ut,N (i 1)
i
<
2
+( (
1 2
ut,N (i)
2
+
1 3
ut,N (i) 3 ) + 3
2
ut,N (i) 2 )
((
1 2
ut,N (0) ut,N
j
+
1 ut,N (0) 3 ) + ut,N (0) 2 ) 3 3 2 3 4 ut,N (i) 2 9 t + max ut,N (j)
2
+
j
+M 3 max ut,N (j)
ut,N (i) 2 t + 2 31
fi,N
2
t + O(t).
We are now left on the right hand side the terms which are bounded due to 4.2.2. Substituting the bounds, we get,
n
(t)1 ut,N (i) ut,N (i 1)
i
2
1 1 ut,N (j) 2 + ut,N (j) 3 ) + ut,N (j) 2 ) 3 2 3 4 1 1 ( ( ut,N (0) 2 + ut,N (0) 3 ) + ut,N (0) 2 ) 3 2 3 4 1 6 2 Csp + C0,f 1 + M 3 C0,f 1 + 2 fi,N 2 t + O(t). +( (
4.4
Existence and convergence to NV
Having shown the existence for 4.2, we would like to show that some subsequence of ut,N (i) converges to a solution to 4.1. We have shown that ut,N (i) W 1,9 H 2 for each i. We will construct such a subsequence by successively reducing the sequence to a convergent sequence that converges at each of the time steps up to the current iteration. We start this process at i = 1. By compactness there exists ut (1) W 1,9 H 2 and a subsequence such that ut,Nk (1) converges to ut (1) strongly in W 1,3 and weakly in H 2 . We now again take the subsequence of ut,Nk to obtain ut (2) such that this sub subsequence converges to it. This process will continue for all i = 1 . . . n. In this way, we can obtain a subsequence ut,Nl and a function ut {i = 1, . . . n} T3 R such that for each discrete time steps i = 1 . . . n, ut,Nl (i) converges to ut (i) strongly in W 1,3 and weakly in H 2 . It remains to show that ut (i) each satisfy 4.1. This is done in almost exactly the same manner as was done in chapter 3. Thus we have, Theorem 4.4.1 There exists a unique solution ut (i), i = 1, . . . n to N Vt ( , 3, M ). Can we take a sequence of problems N Vt as t 0 and conclude that such a sequence converges to a function that satises N V ? To do this, we need to clarify a few things. First, ut (i) is dened on a discrete grid and therefore not suitable when we discuss about its convergence to a function that is dened continuously in time. We must interpolate this sequence between the discrete timesteps to derive a function that is dened on space-time. Secondly, we should decide on the most convenient way by which t converges to zero. 32
To this end, we consider successively rening the mesh. That is, at the kth step, we take t = 2k , so that the mesh at the kth step is a renement of the mesh at the k 1th step. Then, to obtain an appropriate function from the sequence ut (i) so that we can talk about it as a function in time, we interpolate ut (i)s to dene: ut (t) = Notice that for any norm, ut (t) ut (i) + ut (i 1) . This implies for instance that ( ut p ) p
1 <
t ti1 ti t ut (i) + ut (i 1). t t
(
i
ut (i) p t) p ,
1
thus, ut (t) L2 (I; H 2 ) L3 (I; W 1,9 ) L (I; H 1 ). and that t ut (t)
2
=
(t)1 ut (i) ut (i 1) 2 .
thus due to 4.3.2, ut L2 (I; L2 ) and the bound is uniform in k. Thus, by Aubin-Lions compactness theorem, there exists a subsequence that converges to a function u strongly in L2 (I; W 1,3 ). This convergence also takes place for almost all time in I say J I. Take a time t J {j2k j relatively prime to k}, in another words, t is a dyadic time that belongs to J. Now, notice that t ut (t) = whenever t (ti1 , ti ]. We see, therefore that ut almost satises N V at time t, except that the nonlinearity has the dependence on time ti1 . However, due to the convergence of L2 norm of ut (i) ut (i 1) to zero as t 0, the residual term in the nonlinear term goes to zero. Since ut (t) satises N V with residual of order O(t) it suces to show that at t u also satises N V . But we know that the strong convergence takes place at t so it can be shown that u satises N V by a similar method as was shown in chapter 3. Theorem 4.4.2 Let ut (i) be sequence of solutions to N Vt where t = 2k . We dene for each k the interpolant: ut=2k (t) = t ti1 ti t ut (i) + ut (i 1). t t 33 (ut (i 1) + ut (i)) , t
It can be shown that a subsequence of ut=2k exists that converges to u L2 (I; H 2 ) L3 (I; W 1,9 ) L (I; H 1 ) and that u satises N V .
4.5
Stability and uniqueness
We will now show the stability estimate to prove uniqueness. The following stability estimate is interesting in that if the power input is nite so that C0,f < for all time, then the two solutions stay boundedly close to each other. Lemma 4.5.1 Let ut (i)1 , ut (i)2 be two solutions to N Vt . Let t satisfy: t(M + Then, u1 (n) u2 (n) t t
2
<
1
) 2 Ctm
1 4
2(M +
1 )( 1 C
0,f )
2
u1 (0) u2 (0) t t
2
+ O(t)
(u1 (0) u2 (0)) 2 . t t
In particular a solution ut to N Vt is unique. Proof Let wi = ut (i)1 ut (i)2 . Then, subtracting the two equations satised by uj (i) j = 1, 2 and testing against wi we t get, 1 wi 2
2
+ t( (
wi
2
+
wi
p p
+
wi 2 ) t|(wi1
u1 (i), wi )| + t
1 wi1,N 2
2
,
where we have used A.2.2. Due to the Gagliardo-Nirenberg inequality, w therefore, |(wi1
< < < 18 5
w
1 3
w
2 3
ut (i), wi )|
9 4 9 4
ut (i)
2
18 5
9 4
wi1
18 5
wi
18 5
ut (i) ut (i) wi 2 2
( wi ( wi
2
18 5 2 3
+ wi1 wi
4 3
)
2 3
+ wi1
wi1
4 3
)
+ wi1 2 2
<
wi 2 2
+ wi1 2 2
9 wi 2 ut (i) 3 + 4 12 9 ut (i) 3 + wi1 2 4 12 (i) 3 + ut wi 2 3 12 ut (i) 3 + wi1 2 . 3 12 34
where we used the Youngs inequality in the last line. For the low-frequency part, noting that Gagliardo-Nirenberg, by u therefore, |(wi1
<
3
u
1 2
u
1 2
,
ut (i), wi )|
ut (i)
3
wi1
3
wi
3
wi1 + wi wi1 ) < wi 2 wi 2 1 ut (i) 2 + 3 2 + wi1 2 1 ut (i) 2 + wi1 2 3 2 < wi 2 1 M ut (i) 2 + wi 2 2 12 2 1 (i) 2 + + wi1 M ut wi1 2 . 2 12 Let Ai = 1 M ut (i)
2
ut (i) 3 ( wi1
+ 2
ut (i)
3 3
Thus, summarising our calculations, we have that |(wi1 u1 (i), wi )| Ai wi t
2
+ Ai wi1
2
+
6
wi
2
+
6
wi1
2
Then, due to 4.3.2, Ai ( 2 M + 2 Thus we have, wi
<
1
)Ctm
2
+ t( (
wi
2
+
wi
3 3
1 + 2
2
wi 2 ) wi1 2 .
2Ai t wi + (1 + 2Ai t) wi1
1 + t 2
or, wi
2
+ (1 2Ai t)1 t( (
wi
2
+
wi
3 3
1 + 2
wi 2 ) wi1 2 .
(1 + 2Ai t)(1 2Ai t)1 wi1
2
1 + (1 2Ai t)1 t 2
35
Since 1 2Ai t wj
j 2 j
1 2
by hypothesis, we have due to 4.2.1,
+
j
(
i=1 l=i+1
(1 + 2Ai t)(1 2Ai t)1 )(1 2Ai t)2t( (
2
wi 2
2
+
2
wi
3 3
1 + 2
wi 2 )
(
l=1 j
(1 + 2Ai t)(1 2Ai t)1 ) w0
j
+ (1 2A1 t)1 2
w0
+
i=1
(
1 (1 + 2Ai t)(1 2Ai t)1 )(1 2Ai t)2t 2 l=i+1
wi1 2 ). (4.4)
Note then that for tAi 1 , (1 + 2tAi )(1 2tAi )1 22tAi . The lemma follows from 4 this, together with the 4.2.2.
36
CHAPTER 5 ERROR RATE ESTIMATE
In this chapter, we consider a solution to the 3D NSE, and the solution to N V ( , p, M ), and bound the norm of their dierence in terms of the dierence of the initial datum and the regularity of the solution to NSE. As was discussed in chapter 3, we will discuss the error rate in terms of . There, we have indicated that M is allowed to depend on when and go to innity 0. One question is to nd whether there exists an optimal choice of M ( ). The
estimation of the convergence rate oers one possibility to choose M ( ), as we will see that the optimization of the error estimate naturally gives us how M should depend on certain inverse polynomial of . We also note the role played by the parameter p. On the one hand, raising p stabilizes the system; therefore, in the presence of sucient regularity, we obtain a better rate. On the other, higher p implies that N V is a signicantly perturbed version of NSE, and therefore if NSE solution does not have the required regularity, the rate does not apply. We also prove as was done in the previous chapter that the uniform in time rate estimate is possible for p =
5 2 5 2
although the rate now depends on the exponential power of
. In this way, we have indicated how M should depend on and how p should be chosen,
eectively restricting the parameter choices.
5.1
5.1.1
Nonlinear viscosity case
5 2
Small viscosity case
as we have made in the last section. In this section,
We rst make an assumption that p
unlike the stability estimate obtained for the implicit scheme, we would like to obtain an estimate that is completely independent of . The price to pay is that we no longer have an uniform in time error bound. This estimate, however, gives us an insight into the nature of the trade-o between the cut-o frequency M and the articial viscosity coecient . 37
Let u1 be a strong solution to NSE, and u2 be a solution to N V ( , p, M ) such that their initial condition agree: u1 (0, x) = u2 (0, x). We would like to derive an estimate on w = u1 u2 . We know that u1 satises t u1 u1 + P (u1 while u2 satises t u2 u2 P QM Then, we can see that w satises t w w P QM +P (u1 u1 u2 ((1 + | u1 |p2 ) u1 ((1 + | u2 |p2 ) u2 ) u2 ) = P Q M ((1 + | u1 |p2 ) u1 . ((1 + | u2 |p2 ) u2 ), +P (u2 u2 ) = f. u1 ) = f,
We test this equation against w to get, 1 t w 2 + w 2 + ( 2 ( u2 | w + u2 C ( u1
2
w
p1 p
2
+
w p) p u2 , w) w p ) + |(w p u2 , w)|.
w p ) (w ( w
2
+
u1 p ) + p
12
+
We estimate the convective part due to the high-frequency as (w 12 u2 , w) < w
2
u2 p ( w
3 2p3
2
2p p1
+ w w
2
2p p1
)
3
+C
u2
p
2p 2p3
2
+ M 2p
u2
p
w 2,
where we have used A.1.7. For the low-frequency part, (w Consequently, t w
2
u2 , w) M 2 u2
5
w 2.
+ u1
w
2
2
+ (
w
2
+
w p) p u2
p
2p 2p3
C1 (
+
u1 p ) + C( p
3 2p3
+ M 2 u2 + M 2p
5
3
u2 p ) w 2 .
Therefore, letting f (t) = C(
3 2p3
u2
p
2p 2p3
+ M 2 u2 + M 2p
5
3
u2 p )
38
we have t (ef (t) w 2 ) + ef (t) +ef (t) (
<
w
2
w u1
2
+
2
w p) p u1 p ). p
ef (t) (
+
We have that
t 0
3 2p3
u2
4
p
2p 2p3
+ M 2 u2 + M 2p
3 1 p 1 p C0,f T
5
3
u2
p
5 2p3 5
2p3 C0,f T 2p3 + M 2p T
2p5
+M 2 T C0,f . It can be seen that by choosing M ( ) = the same order dependence on c(f, u0 ) max{t, t
2p5 2p3 2 2p3
the terms can be balanced to have
1
.
Therefore, the above expression is bounded by
} where c(f, u0 ) is a constant that depends on the initial condition and
the forcing. Thus, (u1 u2 )(t) e
5 2p3
2
2p5
t
c(f,u0 ) max{t,t 2p3 } 0
(
u1
2
+
u1 p )dt. p
Therefore, we can summarize our result in the following. Theorem 5.1.1 Let u1 be a strong solution to the 3D NSE such that u1 Lp ((0, T ); W 1,p ), and u2 be a solution to N V ( , p,
2 2p3
), such that their initial conditions agree and the
forcing f L1 ([0, T ] : L2 ). Then the following estimate holds for t T , (u1 u2 )(t) e
5 2p3
2
2p5
t
c(f,u0 ) max{t,t 2p3 } 0
(
u1
2
+
u1 p ) dt, p
Note that the error estimate depends on the Lp in time of the solution. Therefore, the error rate for the nonlinear viscosity is sensitive to the singularity that may develop in time. This is intuitive since the nonlinearity should in principle regularize such a singularity and hence keep the turbulence model away from the singular NSE solution. We also note that a larger p also implies smaller power dependence on
1
. Thus, for smooth NSE solution, the stability
of a larger p implies that the N V solution stays closer to the NSE solution. 39
Another notable byproduct of the estimate is that the optimization of the estimate gave us the best value for M . The result is that larger p means we can choose a smaller cut-o frequency. This is consistent with the fact that when the solution is smooth, we can use a more stable scheme.
5.1.2
Uniform in time estimate
We note in passing that as in the last section, we can obtain a uniform in time error estimate
5 for p = 2 . The cost is that the estimate now depends on .
Now, if u1 is a solution to NSE and u2 solution to N V , we have for w = u1 u2 , t w(t) (
2
+ (
2
w
p p
+
w 2)
w
+
w p ) + |(w, u2 , w)| p
Notice that letting p = 5 , we can bound the nonlinearity as 2 |(w, u2 , w)| Due to A.1.7, |(w, u2 , w)| u2
3
u2 w
6 5
5 2
w
2
10 3 3 2 5 2 5 2
u2
3
w
4 5
u2
w
2
+
12
w
2
u2
3
w
2 3
w
w 1 M
u2
2 2
w
2
+
12
w
2
We see that the energy estimate gives
t
1 M
0
u2 w(t)
2 2
+
3 2
u2
5 2 5 2
2 2 M C0,f + 2
3
1
2 C0,f
2
e(
3 2 M C 2 + 2 1 C 2 ) 0,f 0,f
w(0) 2 .
5 Theorem 5.1.2 Let p = 2 , if u1 is a solution to NSE, u1 Lp ((0, T ); W 1,p ) and u2 solution
to N V such that their initial condition agree and that the forcing f L1 ([0, ), L2 ). Then the following uniform estimate holds: w(t)
2
e(
3 2 M C 2 + 2 1 C 2 ) 0,f 0,f
t
(
0
u1
2
+
u1 p ) dt, p
40
The above theorem states that as long as the solution to NSE is smooth and its total uctuation is bounded in a certain appropriate time-space norm, then the error between the NSE solution and N V solution is bounded in time. Thus, they will stay within a tube of constant radius about the origin.
5.2
Hyperviscosity case
In this section we will estimate the error for the hyperviscosity case. Let u1 be a strong solution to NSE, and u2 be a solution to HV ( , , M ) such that their initial condition agree: u1 (0, x) = u2 (0, x). We would like to derive an estimate on w = u1 u2 . We have that u1 satises t u1 u1 + P (u1 while u2 satises, t u2 u2 () u2 . + P (u2 Then, we can see that w satises t w w () w +P (u1 u1 u2 u2 ) = () u1 . u2 ) = f u1 ) = f,
We test this equation against w to get 1 t w 2 + w 2 + | | w 2 2 | | u1 | | w (w u1 , w) C | | u1
2
+
12
| | w
2
+ |(w
u1 , w)|.
First note that due to Gagliardo-Nirenberg, and w for =
43 . 4 4
w = 0 we have
1
w
| | w
,
Thus, u1 | | w
2(1)
(w 12
u1 , w) | | w
2
+ ( M 2 +
3 43
| | w 12 4 u1 43 ) w 2 , w 41
2
2
+C
(1)/
u1
1/
w
2
where we have used A.1.7 and A.1.8. We can also bound the nonlinearity using the physical viscosity term as the regularizing eect as done in the previous section. Consequently, t w
2
+
w
2
2
+
| | w
2
3 43
C1 | | u1 Alternatively, t (e
Rt
0(
+ C2 ( M 2 +
u1
4 43
) w 2.
u1
4,
M 2 +
3 43
u1
4 43 ) ds
w 2) +
w
2
+
| | w
2
C |
| u1 2 .
1
We can again balance the terms so that it can be bounded by a common power of this, we choose M
2 43
. For
. Solving the above equation gives the following estimate: (u1 u2 )(t)
2 u1
4 43 ) ds
e
3 43
Rt
t
0 (1+
|
0
| u1 2 , dt.
In summary, we have the following theorem. Theorem 5.2.1 Let u1 be a strong solution to the 3D NSE such that u1 L2 ((0, T ); H ) L 43 ((0, T ); H 1 ), and u2 be a solution to HV ( , , agree. Then the following estimate holds: (u1 u2 )(t) e
3 43 4 2 43
), such that their initial conditions
2 u1
4 43 ) ds
Rt
t
0 (1+
|
0
| u1 2 .
1
Note the trade-o at work here. We have reduced the exponential dependence on t cut-o M
2 43
,
which is signicant for small . Also, the balancing of the term allowed us to choose the which is smaller than the nonlinear viscosity method.
42
CHAPTER 6 CONTRACTION IN PHASE SPACE
It is known that for the 2-dimensional Navier-Stokes equations, there is a frequency scale beyond which the molecular dissipation becomes dominant and the exponential contraction of the phase space results. In this section, we show that the N V model and HV model in 3D also possess such a characteristic frequency scale when we take the cut-o frequencies to exceed such a scale. In this section, we will estimate the dimension of this nite-dimensional attractor for N V and HV models in 3D. We will assume that the forcing is bounded (i.e. f L ([0, ), L2 )); however, we do not assume that the total energy input is bounded.
6.1
Contraction for the N V model at p =
5 2
We obtain the phase space contraction estimate for the N V model. This is essentially done by considering two distinct solutions to the high-frequency part of N V problem that agree in their low-frequency parts. Because dissipation acts much more strongly on high-frequency modes, we can show that such solutions converge to each other exponentially fast in time. This implies that the dynamics of the equation is mainly concentrated to low modes. Note that lowering the dimension of this exponential attractor without aecting the large-scale dynamics too much is exactly the goal of turbulence modeling. We will see that for N V model, the exponential attractor dimension can be obtained for p=
5 2
because of the same reason that we were able to obtain uniform in time bounds in the
3 2
previous chapters. We will see that the attractor dimension is bounded by
2 .
1
We set the low frequency part: PM u = u and the high-frequency part: QM u = u. Then, t u + PM P (u u u) = 0, (6.1)
43
and t u ( + ) QM P u (| u|p1 u) + QM P (u u) = 0. (6.2)
We now focus on the high-frequency equation. Let us assume that there is a function v that solves the following equation. t v ( + )v QM P (| v|p1 v) + QM P (v v+u v+v u+u u) = 0.
Subtracting, we have the following equation for w = u v , t w(+ )w QM P (| u|p1 u| v|p1 v)+QM P ( uv v+ w+w u) = 0. u u Now, we test this equation by w to obtain, 1 t w 2
2
+ ( + )
w
2
+
w
p p
+ (w
u, w) + (w
u, w) 0.
Now, due to the fact that w lives in the high-frequency space, we have w M w . Then, notice that (w
<
u, w) u
5 2
u w
5 2
w
6 5 <
10 3
w
4 5
w 2 ( + ) 2
3
u
5 2 5 2
+
+ 12
w
2
Therefore, we get 1 t w 2 + ( + ) w 2 2 3 + w 2 + (( + ) 2 2 Therefore, 1 t w 2 Now we let f (t) = 2(( + )M 2 We know that t
1 0 2
u
5 2 5 2
+ ( + )1
u 2) w 2. 3
(( + )M 2 2( + ) 2
3
u
5 2 5 2
( + )1 M
u 2) w 2.
1 t
t 0 t
(( + ) 2
3
u
5 2 5 2
+ M ( + )1
u 2 ) ds.
u
5 2 5 2
1
2 Cf,,0 ,
44
where
2 Cf,,0 = u0 2
+
1 f
2 L ([0,),L2 ) .
Therefore, f (t) 2( + )M 2 (( + ) 2 Using the fact that Cf,,0 2 , we see 2( + )M 2 ( + ) 2 ( )1 M ( + )1 2 Thus we may take ( + )2 2 so that there exists a > 0 such that f (t) . Then, t (ef (t)t w 2 ) 0. Therefore,we can summarize what we found as follows: Theorem 6.1.1 Let u solve N V ( , 5 , M ) and v solve the high-frequency equation for 2 N V ( , p, M ) with low-frequency forcing by u, then we have the following contraction estimate: u(t) v(t) 2 (t) et u0 v0 2 , when ( + )2 2 C < M, for suciently large constant C. (6.3)
< 3 < 1 3
1
2 2 Cf,,0 + M ( + )1 1 Cf,,0 ).
f (t).
M,
6.2
Attractor for HV with =
3 2
3 2
We now consider the hyperviscous model for =
and show that in this case, we can
obtain the contraction estimate even in 3D, and therefore can estimate the dimension of the attractor. The high-frequency part has the following form: t u () u + QM P (u u 45 u) = 0.
Let us assume that there is a function v that solves the above where v(0) = u and with the low-frequency part u used as the forcing. In other words, t v v () v + QM P (v v+u v+v u+u u) = 0.
Notice that if we took PM of the above equation, v satises t v v () v = 0 . Since v (0) = 0, v (t) = 0 for all time. Thus, v consists only of the high-frequency part. Subtracting, we have the following equation for w = u v , t w w () w + QM P ( u We test the equation by w to obtain, 1 t w(t) 2 Note that w Since
3
6 52
uv
v+u
w+w
u) = 0.
2
+
| | w
2
+ (w
u, w) = 0.
| | w 2 .
+
52 6
5 = 6,
w we get (w but we have, w where u, w) = (w
w
6 5
w
3
w
6 52
w, u) < u
6
w
w
6 < 5
u
6
w
3
| | w 2 ,
3 <
w
2
| | w
1
.
= and hence 3 . Thus, 2 u
6
4 3 . 2
w
3
| | w
2
2
<
u
6
w
| | w
2
2
u
6
w
2
+
| | w 2 .
Now, note that due to the Poincare(forward) inequality and that w Ran(QM ) we have M w
<
| | w .
46
Consequently, t w(t)
2
<
( M 2 2 u
4 43
6
3 43
) w 2.
Then, to respect the energy dissipation we must set =
t
3 2
so that u 2 )1/2 . 2
(t1
0
u
4 43
6
)
43 4 <
t
(t1
0
Summarizing, Theorem 6.2.1 Let u solve HV ( , 3 , M ) in n = 3, and v solve the high-frequency equation 2 for HV ( , , M ) with low-frequency forcing by u, then we have the following contraction estimate: u(t) v(t) 2 (t) et u0 v0 2 , where M > C( 2 )1/3 . for suciently large C. Note the small dependence of M on power of . We can see from this that the dimension of the exponential attractor is rather manageable. For more information about attractors and its nite dimensionality in the setting of hyperviscosity, see [35, Temam]. (6.4)
47
CHAPTER 7 INTRODUCTION TO INFINITE PRANDTL-NUMBER EQUATION
Consider the generalized non-dimensional Boussinesq model: (for instance, see [38, Turcotte]) 1 (t u + u Pr u = 0, t u) (() u) + p = Rak,
(k() ) + u
= 0,
(7.1)
on the domain = T2 [0, 1]. The temperature satises the Dirichlet boundary condition: (x, y, 0) = 1, (x, y, 1) = 0, while velocity satises the free-slip condition [38, Turcott]: u3 (x, y, 1) = u3 (x, y, 0) = 0, 3 u1 (x, y, 1) = 3 u1 (x, y, 0) = 0, 3 u2 (x, y, 1) = 3 u2 (x, y, 0) = 0 We use the following convention for labeling the variables x = (x1 , . . . , xn ) = (x , xn ). P r is the Prandtl number and Ra is the Rayleigh number. Notice that the viscosity and the conductivity k depend on the temperature. Such dependence can model various physical phenomena especially in geophysical applications. In this work, we only consider a very viscous ow in which the inertial term can be neglected. The notable physical example of such a ow is the mantle ow in the interior of
48
the Earth. For simplicity, we also neglect the temperature dependence of the conductivity: (() u) + u = 0, t + u = 0. (7.2) p = Rak,
In the mantle ow, the viscosity depends strongly on the temperature. For example, a typical function that is used in the geophysics literature is () = exp(a||) [38, Turcotte pg. 319], where a quanties the viscosity contrast, e.g. = exp(a) = exp(5log(10)) = 105 . The temperature independent viscosity case has been analyzed extensively; for instance by [3, Constantin and Doering]. A rather detailed analysis is possible in that case since the Stokes equation is linear and hence techniques from potential theory can be used. In fact, in the constant viscosity setting, the real challenge is the estimation of a constant called the Nusselt number which quanties the ratio of heat transported due to the convection and that due to the conduction and is dened as follows: 1 1 T N u = lim sup || | |2 . T T 0 Since convection is a nonlinear transport process in our equation, the Nusselt number also describes the degree of nonlinearity present in our equation and is typically of order some power of Ra. Estimating the exact power occupies a large part of the current interests of the researchers in this area. We would like to aim for such an extensive analysis for our equation. However, there is a signicant mathematical challenge in our case, since the viscosity depends on the temperature which introduces a second order nonlinearity. On top of this, unlike the well-studied second order nonlinearity such as the p-Laplacian, what we have here is not monotonic in the temperature variable. Therefore, a variation in the temperature can potentially cause a large variation in the velocity eld. Nevertheless, the regularity of the solution to this equation, and therefore the well-posedness can be shown. We can not, however, be satised with just a regularity result. What matters here is the actual constants in the estimation of the regularity of the solutions as these constants can be used to bound the Nusselt number. Before plunging into the regularity, we note that the dening feature of the ow we study is the boundary layer behavior. In the next section, we will examine a simplied onedimensional model from which we obtain some insights into the behavior of the boundary 49
layer. We learn that the velocity has a strong decay toward the surface, eectively modeling the transition from the upper mantle layer to the lithosphere. Various estimates are expressed in terms of the important parameters and Ra.
7.1
One dimensional model
In the mantle, the viscosity is known to depend strongly on the temperature. This has an important eect on the dynamics of the ow since it implies a formation of a stagnant layer at the surface that models the transition from the hot and uid upper mantle to the cool and elastic lithosphere. The mathematical challenge is to predict such a behavior rigorously from the equation. We will rst analyze an articially simplied model in 1D:
x (()x u) + x p = Ra, t xx + ux = 0, where p is given, (0) = 1, (1) = 0, u(0) = u(1) = 0, and |t=0 = 0 . (7.3)
7.1.1
Boundary Decay Estimate by Comparison Principle
In what follows we assume 0 0 1, and hence by the maximum principle the temperature lies between 0 and 1. We start with the velocity equation: x (()x u) + x p = Ra, with u(0) = 0, u(1) = 0. The goal is to understand the eect of temperature dependent viscosity in the boundary layer. To do so, we dominate the solution from above and below by carefully chosen super(sub)solutions that clarify the role of () on the boundedness of u. The consequence is that we obtain an estimate on the Lipshitz norm of u, and the boundedness of u in terms of ()1 . Lemma 7.1.1 We have for solution u of 7.4
x0
(7.4)
|u(x) u(x0 )| C(Ra + max |x p|)
x
((s))1 s .
50
Proof Let x0 (0, 1), u0 = u(x0 ) and consider Rx 0 ((s))1 s +()Al Rx 0 ((s))1 s + u0 , x x0 x 0 Rx , ux0 ,+() (x) = ((s))1 s (+)Ar Rx0 + u0 x > x0 1 1 s ((s))
x0
where Al = max{(Ra + max |x p|) Ar = max{(Ra + max |x p|)
x0 0 1 x0
((s))1 s, |u0 |} . ((s))1 s, |u0 |}
We claim that ux0 ,+() is a super(sub)solution to 7.3 at x0 . For the supersolution, we plug into the equation for x < x0 to get x (() Al (()1 x)) + x p = Al ( x0 1 s ((s)) 0
x0 0
((s))s)1 + x p Ra Ra,
by maximum principle. Also, ux0 ,+ (x0 ) = u0 and ux0 ,+ (0) = Al + u0 0. Other cases follow similarly. Thus, due to the comparison principle for linear elliptic equation [13, Gilbarg and Trudinger, pg.268-270], ux0 , u(x) ux0 ,+ . Suppose x0 < 1/2, we can then bound u(x0 ) by u0,+() , a barrier function at x = 0 |u(x0 )|
x0 ((s))1 s 0 Ar 1 ((s))1 s 0 x0
c(Ra + max |x p|)
0
((s))1 s.
Thus, we see that for x < x0 ,
x0
u(x) u(x0 ) Al (
x
()1 s)(
0 x0
x0
()1 s)1
x0 x
max{(Ra + max |x p|)
0 x0
((s))1 s, |u0 |}(
()1 s)(
x0 0 x0 0
()1 s)1
max{(Ra + max |x p|)
0 x0
((s))1 s, c(Ra + max |x p|)
((s))1 s}
(
x
()1 s)(
0
x0
()1 s)1
x0
C(Ra + max |x p|)
x
()1 s.
1 x0
The case x > x0 is easier since
()1 s does not become singular as x0 1/2. The
same reasoning works for x0 > 1/2 by using the barrier function at x = 1 instead.
51
Notice that since super(sub)solution depends on the temperature, we expect that by establishing a good bound on the temperature at the surface, we can expect a better than a linear decay there. To do this, we focus on bounding the temperature: t xx + ux = 0. We establish a supersolution. Lemma 7.1.2 We have for solution of 7.5,
1
(7.5)
(x) Proof Note that
C(Ra + max |x p|)
x
eC(Ra+max |x p|)(1t) dt.
2
()1 ea hence, due to 7.1.1, |u| C(Ra + max |x p|)(1 x). We use an error function, a useful tool in the theory of boundary layers.
1
f (x) =
C(Ra + max |x p|)
x
eC(Ra+max |x p|)(1t) dt,
2
we claim that f is a supersolution of 7.5. Note that f is a supersolution if t f xx f C(Ra + max |x p|)(1 x)|x f | |u||x f |, which is equivalent to the condition, with the explicit ansatz, that 2(C(Ra + max |x p|))3/2 (1 x) C(Ra + max |x p|)(1 x)(C(Ra + max |x p|))1/2 , and if f (0) > (0) = 1. Thus, the claim follows. Consequently, we can bootstrap the bound on the velocity, Lemma 7.1.3
1
e(a
C(Ra+max|x p|)(1s))
|u(x)| C(Ra + max |x p|)
x
s ds.
Proof Note that due to 7.1.2, ()1 = ea e(a
C(Ra+max|x p|)(1x))
Notice that the function on the right hand side shows a strongly decaying prole at x = 1, illustrating the strong decay of heat transport in the vertical direction at the surface. 52
CHAPTER 8 WELL-POSEDNESS OF INFINITE PRANDTL MODEL
In this section, we show the well-posedness for the given equation. The well-posedness can be proved in many ways. We will rst give a priori estimates that guarantee the existence of a weak solution. Then, in successive steps, we improve the regularity of the solution until the uniqueness can be shown. Here we consider the following system of convective heat equation coupled with Stokes system: (() u) + u = 0. p = kRa, (8.1)
t + u
= 0.
(8.2)
on the domain = T2 [0, 1]. The temperature satises the Dirichlet boundary condition: (x, y, 1) = 0, (x, y, 0) = 1, while velocity satises the free-slip condition: u3 (x, y, 1) = u3 (x, y, 0) = 0, 3 u1 (x, y, 1) = 3 u1 (x, y, 0) = 0, 3 u2 (x, y, 1) = 3 u2 (x, y, 0) = 0 We assume that 1 .
53
In particular we will be especially interested in expressing the various estimates in terms of two key quantities: Ra and . Let w = (1 x3 ) be a perturbative variable. Then our equation can be re-expressed in terms of w: ((w + (1 x3 )) u) + u = 0. p = kRa(w + (1 x3 )), (8.3)
t w w + u with w = 0 on .
w u3 = 0.
(8.4)
8.1
Extension to the perodic domain
In this section we will show that due to the free-slip boundary condition, a solution to 8.3 and 8.4 can be extended to the periodic domain and satisfy a slightly modied version of the original equation. On , this new version reduces to the original. We extend u and w = (1 x3 ) to T2 [1, 0] in the following way: w(x1 , x2 , x3 ) = w(x1 , x2 , x3 ), u3 (x1 , x2 , x3 ) = u3 (x1 , x2 , x3 ), u2 (x1 , x2 , x3 ) = u2 (x1 , x2 , x3 ), u1 (x1 , x2 , x3 ) = u1 (x1 , x2 , x3 ), p(x1 , x2 , x3 ) = p(x1 , x2 , x3 ). We then identify the two ends of [1, 1] to complete the periodic extension. In other words, we make odd extensions for w and u3 and even extensions for u1 , u2 and p. This means that if w and u3 are smooth, then they have up to the rst derivative that is continuous across the boundary points x3 Z due to the Dirichlet boundary conditions. For u1 and u2 , they are continuous on Z due to the even extension. Since their normal derivative is zero, we also have that they are continuous up to the rst derivatives. Note that the kth order normal derivative of an odd(even) function at a point x0 has the same(dierent) sign as its value at the point x0 if k is odd(even), and the sign ips if k is even(odd). It can be seen that then, u and w satisfy the following on T2 [1, 0]. For ease 54
of notation we only express the variable x3 [1, 0]. t w(x3 ) w(x3 ) + u(x3 ) = t w(x3 ) + w(x3 ) u1 (x3 )1 w(x3 ) u2 (x3 )2 w(x3 ) u3 (x3 )3 w(x3 ) +u3 (x3 ) = 0. Thus, the extended function satisfy the original heat equation for x3 [1, 0]. To extend , we must extend 1 x3 in the following way: g(x3 ) = 1 x3 1 x3 0 x3 1 1 x3 1 x3 0, An important remark is that extension of w extends (w + g) in way that it preserves g(x3 ) = Hlder continuity and Lipshitz continuity. Now, let x3 [1, 0] then, o (w + g)(x3 ) u1 (x3 ) = 1 (w + g)(x3 )1 u1 (x3 ) w(x3 ) u3 (x3 )
+2 (w + g)(x3 )2 u1 (x3 ) + (3 (w + g)(x3 ))(3 u1 (x3 )). Also for u2 , while, (w + g)(x3 ) u3 (x3 ) = 1 (w + g)(x3 )(1 u3 (x3 ))
+2 (w + g))(x3 )(2 u3 (x3 )) + (3 (w + g)(x3 ))3 u3 (x3 ). So, for instance for the rst component, (w + g)(x3 ) u1 (x3 ) ((w + g)(x3 ))u1 (x3 ) + 1 p(x3 ) u1 (x3 ) ((w + g)(x3 ))u1 (x3 )
= (w + (1 x3 ))(x3 ) +1 p(x3 ) = 0. while for the third component, (w + g)(x3 ) = (w + g)(x3 )
u3 (x3 ) (w + g(x3 ))u3 (x3 ) + 3 p(x3 ) u1 (x3 ) + (w + g(x3 ))u3 (x3 )
3 p(x3 ) (kRa(w(x3 ) + x3 (x3 )) = 0. Thus, the velocity equation is satised. For incompressibility, the signs of 1 u3 and 2 u2 do not change when the point is reected due to their evenness, while 3 u3 also has the same sign since the normal derivative do not change sign by reection. 55
8.2
showing existence. Heres our rst a priori estimate. Lemma 8.2.1 If u solves 8.1, then
Existence
In this section, we obtain various estimates for the equations 8.1 and 8.2, with a goal for
1/2
| u|2
c 1 Ra
L2 .
Proof We test the Stokes system by u to get | u|2
L2
()| u|2 Ra
x3 1/2
u3
Ra cRa
(
0
3 u3 )2
1/2
L2
|3 u3 |
2
.
We now compute the standard a priori estimates. Lemma 8.2.2 t 2 + | |2 C 1 Ra.
Proof Set w = (z) for to be chosen later. We get that w satises t w w + u3 + u Testing against w, we have 1 t w2 + 2 Thus by Cauchy-Schwartz, t w2 + but | |2 = = | w + k|2 3 w + ( )2 . | w|2 2 |u3 w| + ( )2 , u3 w + | w|2 + 3 w = 0. w = 0.
| w|2 + 2 56
By adding two times the rst equation with the second, we get, t (w2 ) + 2 = ( )2 . u3 w + (| w|2 + | |2 )
We now choose a specic background prole C ([0, 1]) where (x3 ) = and 1 (x3 ) . when 1 x3 1. We then bound the following term: u3 w c Ra c 1 Ra(
1x3 1 1 x3 L2
1 0 x3 1 0 x3 = 1,
1
1
1
3 w
1x3 1 x3 x3
3 u3
1
(3 w)2
1x3 1 1 1/2
(3 u3 )2
1 1/2
|3 u3 | w
2 L2
2 1x3 1 x3
(3 w)
2
L2
+
w
2 L2 ),
where Poincar works for since it vanishes at xn = 0. We then choose c 1 Ra = 1/4, so e that t (w2 ) + 1 2 (| w|2 + | |2 ) C 1 Ra.
Lemma 8.2.3 t
L2 (0,T ;H 1 )
L2 (0,T ;L2 )
+ C Du
L (0,T ;L2 )
L2 (0,T ;H 1 ) .
57
1 Proof Next we obtain the time derivative estimate for . Let H0 ().
t = Integrating in time, t | |2
1/2
+
u
1/2 1/2
+
1/2
(u)2 |u|
6
1 3
| |2
1/3 1/2
| |
2
+
1/2
||3
1/2
| |2
1/2
| |2
+C
|Du|2
H1
| |2
.
L2 (0,T ;L2 )
+ C Du
L (0,T ;L2 )
L2 (0,T ;H 1 )
L2 (0,T ;L2 ) .
We now use the standard Galerkin method in which we solve the following discrete
1 system. Let {k } be dense in H0 () and {vk } be dense in (H 1 ())n {v :
v = 0}. Let
HN = span{k }N , VN = span{vk }N . 1 1 We also modify as follows: c () = ea min{||,1} The approximate problem is as follows: Find (uN , N ) VN HN such that c (N ) uN : t N j + vj = N RaN vj,3 j + uN N j = 0, (8.5) for all (vj , j ) VN HN . The existence of a unique solution to this ODE follows from the following facts. The rst equation for a xed temperature admits a unique solution because Stokes equation on the divergence-free space is symmetric and positive denite. Also, if 1 and 2 are two data to the rst equation, then the dierence w = u1 u2 satises: w
2
c (1 )| w|2
<
(1 ) (2 ) 58
2
u2
2 2
+ Ra2 1 2 2 . 2
But due to the equivalence of norms in nite-dimensional vector space, this implies that the solution map of the rst equation is Lipshitz. Thus, we can express the above system as a single equation by representing uN in the second equation in terms of the temperature using the solution map of the rst equation. The local existence now follows from the Picard theorem using the fact that the nonlinearity is locally Lipshitz. The a-priori estimates are valid for the approximate problem as well, and hence we have 1. N L (0, T ; L2 ) L2 (0, T, H 1 ), 2. t N L2 (0, T ; H 1 ), 3. uN L (0, T ; H 1 ). This implies that the local solution is in fact global. Now, we can use Aubin-Lions compactness theorem to obtain a subsequence {Ni } converging weakly in L2 (H 1 ), and strongly in Lp (L2 ) to a function for some p. We also have that uNi converging to u weakly in Ll (H 1 ) for any l < . Clearly, for the linear terms, the weak convergence suces to gurantee the convergence of the inner product in the Galerkin formulation to the corresponding one in the weak formulation. For nonlinear terms, notice that for any sequence ai bi , if ai a strongly in L2 and bi b weakly in L2 then, convergence of ai b i ab. Thus, strong convergence of N and weak uN implies convergence of the form c (N ) uN : vj
while the strong convergence of uN and weak convergence of N implies the convergence of the convective term. In this way the limit can be taken in each terms of the approximate problem and the limit satises the weak formulation. Thus we have proved the following theorem. Theorem 8.2.4 There exists (u, ) such that, 1. L (0, T ; L2 ) L2 (0, T, H 1 ), 2. t L2 (0, T ; H 1 ), 3. u L (0, T ; H 1 ). 59
and satises the weak formulation: c () u : t + v= Rav3 +u = 0, (8.6)
1 for almost all t and all (v, ) H0 L2 such that
v = 0 and
= 0.
8.3
Maximum Principle Estimate
The Galerkin method does not allow us to take advantage of the maximum principle. However, having constructed a solution, we can now use the maximum principle to improve the regularity. As a side product, we can improve the dependence of L (L2 ) L2 (H 1 ) norms on 1 Ra, which is needed for bounding the Nusselt number. Lemma 8.3.1 minQT (x, t) maxQT for all (x, t) QT . Proof Assume the contrary, and let maxQT l < maxQT . Let v = ( l)+ , which by
1 above hypothesis belongs to L2 ((0, T ) : H0 ). We test the equation by v, to get
t ( l)+ ( l)+ + Thus, v = 0 and so the claim follows.
| ( l)+ |2 +
u ( l)+ ( l)+ = 0.
Note that this implies that we can replace c back by in the weak formulation. Lemma 8.3.2 If (u, p) solves 8.1, then
1/2
| u|
2
c 1 Ra,
Proof The velocity bound is similar to 8.2.1, except that we use the L norm bound on the temperature. The maximum principle allows us to improve on the temperature estimate: Lemma 8.3.3 t 2 + | |2 C( 1 Ra)2/3 . 60
Proof Set w, as in the proof of 8.2.2. We then use the maximum principle to get w so that u3 w c c c
1
c,
3 u3
1x3 1 x3
|3 u3 |
1/2
|3 u3 |
2
1/2
c2 1 Ra 1/2 . We then choose c 1 Ra 1/2 = 1 , so that t (w2 ) + 1 2 (| w|2 + | |2 ) C( 1 Ra)2/3 .
The important corollary of the above is that we have the following bound on the Nusselt number. Corollary 8.3.4 N u = lim sup
T
1 1 || T
T
| |2 c( 1 Ra)2/3 .
0
We note that in the case of the temperature independent viscosity, the best bound obtained is due to [3, Doering and Constantin], and is of logarithmic factor times Ra1/3 .
8.4
Stability and Uniqueness
In the case of a temperature independent viscosity, the well-posedness can be shown easily from what has so far been proved. Not necessarily so for the temperature dependent case, as the uctuation of the velocity now depends strongly on the uctuation of the temperature. To see this, we show the following stability result. Lemma 8.4.1 We have that for any (i , ui , pi ), i = 1, 2 solutions to 8.2 and 8.1, the following inequality holds: (1 2 )(t)
2
(1 2 )(0)
2
exp(t(( 1 Ra)2 + (a 1 )2 61
2
u2 4 )).
Proof Let (i , ui , pi ), i = 1, 2 be solutions to 8.2 and 8.1. Let = 1 2 and w = u1 u2 . Then, t + u1 Test against to get, 1 t ||2 + 2 Thus, t ||2 + From the velocity equation we have ((1 ) w) (((2 ) (1 )) u2 ) + (p1 p2 ) = Rak, | |2 |w2 |2 . | |2 = w2 . +w 2 = 0.
We see rst that, |(1 ) (2 )| = |e
a|1 |
e
a|2 |
|2 (x)|
|=|
|1 (x)|
eas ds|a |1 (x) 2 (x)|a
So ||Lip a. Note that due to Gagliardo-Nirenberg and Sobolev inequalities, ||2 |Du|2
2 3
u
2 6
2
u
2
so, testing the dierence of the velocity equations against w, (1 )| w|2
<
Ra
w3 + a2 2 + 4
2 | u2 |2 +
1 4
(1 )| w|2
2
<
1 Ra2
| w3 |2 + 1 a2
u2
2
+
4
(1 )| w|2 .
Therefore, t ||2 + | |2
<
( 1 Ra)2 + (a 1 )2
2
u2
4
2.
The result follows by Gronwalls inequality. Notice the terms in the exponential factor. The presence of the that this term is bounded.
2
u2 is specic to the
temperature dependent viscosity case; and therefore, our aim in the next section is to show
62
8.5
Regularity
In this section we show that a weak solution has a further regularity. We can approach this in many dierent ways. Our hope is that in each estimation process we strive for a bound that is as tight as possible, since such estimates might be used to improve the Nusselt number bound. This goal will not be met by this thesis. Therefore, this remains as one of the future goals of our research. We will rst consider the global regularity of the temperature eld. We will use the Galerkin method. Lemma 8.5.1
t 2
w(t)
2
+
0
eRa
4 (st)
3
w ds eRa t (
4
2
w(0)
2
+ Ra2 )
Proof In the following, we assume that in fact w is an N dimensional Galerkin projection. We take the Laplacian of the Galerkin formulation to get (t w + +u u w
2
w + 2 u :
w
w u3 ) dx = 0
We test against the nite dierence w, we get, 1 t w 2 +
2
+
2
w
2
(2 u :
ww + u
ww u3 w) = 0
Integrating by parts on the nite-dierence, 1 t w 2 + (2
2
+
w
2
( u:
2
w)w u w w u3 w) = 0
+ u:
ww +
63
Integrating by parts, and Cauchy-Schwartz we get 1 t w 2
<
2
+
w
2
| u w|2 + | u w|2 + | u|| 1 2 1 2 | w| w
2
2
w|2 + | u3 |2 dx
+
<
u
2
+
u
2
w
2 4
+
u
2
+
| w|
Also, due to Gagliardo-Nirenberg, w
2
w
1 2
3
w
1 2
,
2
w
2 4
2
w
1 4
3
w
3 4
.
Therefore, 1 t w 2 2 < w
<
2
+
3
w w u
2 2
Ra4
+ 1 2 w 2 + Ra2 + 2
3
u
3
2 2
w
1 2
3
w
3 2
.+
u
2
w
Due to elliptic regularity,
w
<
w
Thus use Gronwall the temperature estimate and the previous lemma:
t 2
w(t)
2
+
0
eRa
4 (st)
3
w ds eRa t (
4
2
w(0)
2
+ Ra2 )
We note that
2
w is remains bounded under the extension since the rst derivative of w
matches at the boundary. Thus, the result follows. Now, we can understand the following regularity result in this setting. Lemma 8.5.2 |
2
u|2
<
((aRa)2
2
w
4
+ ( 1 Ra)2 )
(8.7)
64
Proof Note that since has a matching derivative at the origin, we can take the weak derivative of (). We will again use the Galerkin projection. We take the partial derivative to obtain: c (w + g(x3 )) k u : Testing against k u we get, | k u|2
<
v + k (w + g(x3 )) u :
v=
Rak wv3
|k (w + g)|2 | u|2 + ( 1 Ra)2
|w|2 +
4
|kk u3 |2
Exactly as in the proof of stability 8.4.1, we bound |k (w + g)|2 | u|2 a2 Consequently, | k u|2
<
w
2 6
u
2 3
a2
2
w
2
u
2
u
((a 1 Ra)2
2
w
4
+ ( 1 Ra)2 ) +
2
2
u
2
Combining the estimates for all directions k gives the result. Thus, we have bounded Theorem 8.5.3 Suppose that
2
u , and therefore the uniqueness follows:
2
w(0) < , then the weak solution to the initial value
problem for the innite Prandtl number equation is unique.
65
CHAPTER 9 CONCLUSION AND FUTURE WORK 9.1 Nonlinear viscosity model
In this thesis, we have examined two dierent problems in uid dynamics. The rst was about a particular turbulence model in which an articial spectral viscosity was used to make the simulation of turbulence tractable. The model introduced various parameters and we posed a question whether an eective choice of a parameter can be made using the mathematical analysis. First, to show its basic eectiveness we proved that the resulting partial dierential equation is well-posed. Then, we considered a semi-implicit discretization of the equation, and investigated a particular model in which the forcing dissipates in time. In this case, we were able to prove the uniform-in-time stability of the resulting model for
5 a very specic p = 2 . We have also proved its consistency by showing that it converges to
N V as the time-step goes to zero. To address the question of consistency, we derived an error estimate between the solution to N V and a smooth solution to NSE. The same task was undertaken also for the HV model. We discussed why innity as should be taken to zero instead of M . In fact, we have shown and is taken to
5 2p3
that the convergence to a weak solution to NSE occurs if M depends on us the optimal rate for N V , while we obtained M ( ) =
2 43
0. The error rate gave us an additional insight in that M ( ) =
gives
for HV . For the N V model,
5 2
we have also derived a uniform-in-time estimate for p = 5 . Thus, out of the innity many 2
5 models we have considered parametrized by , p and M , N V ( , 2 ,
) seems to oer a good
choice in terms of the various properties that it is shown to possess.
66
9.1.1
Future work
For the semi-implicit scheme, we left the question of convergence rate estimate as a problem for future work. It is interesting to see if we could obtain a rate that is uniform in time. It is also interesting from turbulence point of view, to try to obtain a local estimate rather than global one. For example, in some cases, we may want to stabilize the system only at those locations in space where the uctuation occurs. One diculty in this direction for our model is that because of the p-Laplacian, the pressure may not have a good local regularity and may cause problem for local estimates.
9.2
Innite Prandtl number model
In the second problem, we considered a model of mantle convection derived from a viscous limit of the Boussinesq model. The novelty in our case was the temperature dependent viscosity that provides a strong coupling between the Stokes equation and the heat equation. In order to understand such a strong coupling, we considered a simple 1D model that exhibits a boundary layer behavior caused by such a coupling. Some intuitions were gained by such a process since maximum principle type arguments were applicable for 1D. For 3D, we approached the question of well-posedness by using the tools from parabolic regularity theory. The local estimates are obtained and well-posedness shown. Unfortunately, these local analysis did not provide understanding for the long-time transport behavior of this equation quantied by the Nusselt number. The investigation into this aspect of the problem will be left for a future research.
67
APPENDIX A MATHEMATICAL BACKGROUND
In this section we will gather a set of mathematical results that are used frequently in our analysis.
A.1
Multiplier theory, lter operator, and fractional dierentiation and integration
The analysis of the spectral viscosity equation involves an interaction of a rather broad range of mathematical concepts. The lter operator is dened in the frequency space, while the nonlinear viscosity involves eects on the physical space. The hyperviscosity brings the issue of the fractional dierentiation and how it interacts with the lter. In this section, we will present a set of results that can serve as a common framework in which these concepts can be manipulated. Due to the use of spectral ltering, it is no surprise that the tools from harmonic analysis will be used extensively. We denote n to be the number of dimensions. First, recall that we dened the operator PN as follows: PN f =
|k| N
f (k)eikx .
Thus, we can consider PN as an operator that is multiplied by |k| N in the frequency space. An operator that is obtained in this manner is called a multiplier operator. For a notational convenience, we denote Tm as an operator obtained by the multiplication by m in the frequency space. Therefore, PN = T|k| N . We will also be interested in how this operator interacts with the fractional dierentiation 68
operator: | |s PN = T|k|s |k| N , and also how I PN interacts with the fractional integration operator: | |s (I PN ) = T|k|s |k| >N , both under which s 0. These operators will become fundamental to the discussion that follows. We need to be able to manipulate these operators analytically; that is, we need estimates in various norms. We call a linear operator T that maps a measure space into another is of type (p, q) if T
Lp Lq
< .
Thus, the goal is to obtain (p, q) estimates for various values of p and q. As we will see later, choosing the cube as the frequency region to which we project: |k| N , is important. We could not have chosen |k|2 there for a rather deep reason in harmonic analysis. We will briey mention this issue later. We can immediately get that | |s PN is an operator of type (2, 2). Let d denote the dimension. Lemma A.1.1 For s 0, | |s PN Proof | |s PN f
2 L2 L2
<
Ns
=
|k| N
|k|s |f |2 ds/2 N s f
2
= ds/2 N s f
2
The (2, 2) estimate is not exible enough for our analysis. We want to obtain a (p, q) type estimates. One convenience for choosing |k| is that the cut-o operator can be expressed in the physical space as a convolution with a Dirichlet kernel: DN (x) =
l
(
sin((N + 1/2)xl ) ), sin(xl /2)
so that, PN (f ) = DN f. 69
Obtaining a (p, q) estimates for each values of p, q may be tedious. The well-known RieszThorin interpolation theorem, allows us to obtain a (p, q) type estimates when (1/p, 1/q) is a convex combination of two end-point types. Thus, we can simplify the task of obtaining an innite number of estimates to just two [1, Bergh and Lofstrom]: Theorem A.1.2 (Riesz-Thorin) Let T be an operator satisfying T and T Let 1 1 1 1 1 1 ( , ) = (1 )( , ) + ( , ), p q p 1 q1 p 2 q2 then we have T
Lp Lq Lp2 Lq2 Lp1 Lq1
< ,
< .
T
1 Lp1 Lq1
T
Lp2 Lq2 .
We will now show that PN
L1 L
and PN
Lp Lp
can be estimated. The desired
estimate follows by an interpolation. First, the former is easy to prove: Lemma A.1.3 PN g
cN d g 1 ,
Proof We will simply estimate the maximum norm of the Dirichlet kernel. Note that if 0 t then, |t|/ | sin(t/2)| and | sin(t/2)| min{|t|, 1} for all t. Therefore, | sin((N + 1/2)t) min{(N + 1/2)|t|, 1} | sin(t/2) t
<
N
by symmetry this holds for all 0 t 2. Thus, DN From which it follows that DN g
<
N d.
DN
g
1
cN d g 1 .
70
The Lp Lp bound is more dicult. Actually, since DN
1
log(N ), we can show such
a bound up to a logarithmic factor by simply using the Youngs inequality. However, the logarithmic dependence can be eliminated. The trade-o is that we need to use the theory of singular integral operators and multiplier theory which implies the non-triviality of such a bound. In fact, it is a result of [10, Feerman] that had we chosen to use a cut-o lter with a square norm:
|k|2 N
u(k), no such bound can exist. However, for the partial sum
operator we use, such a bound is true. The relevant tool to show this has been of historical signicance in the development of harmonic analysis. The tool to use is to show that the question about the boundedness in Lp 1 < p < of operator on a torus can be reduced to the corresponding question on Rd due to the following transference principle which is proved in [19, Krantz]: Theorem A.1.4 (Transference Principle) Let Tm be an operator associated with a multiplier m : Rd C such that m is continuous at each point of Zd , then the restriction m = m|Zd denes a multiplier operator on L2 (Td ), and, Tm
Lp (Td )Lp (Td )
Tm
Lp (Rd )Lp (Rd ) .
Thus, we can transfer the question about a multiplier operator on a torus to a corresponding question on Rn . The boundedness of the multiplier operator on Rd is answered by another fundamental theorem in harmonic analysis (see [1, Bergh and Lofstrom]): Theorem A.1.5 (Hormander-Mikhlin) Let m : Rd C satisfy the homogeneous symbol estimates of order 0: |
k
m()| < ||k ,
Lp (Rd )Lp (Rd )
<
for all = 0 and 0 k d + 2. Then, Tm
cp for all 1 < p < .
We will not give the detail of the proof, but using this principle, the boundedness of PN in Lp follows by noting that it can be expressed as a combination of modulation and a Hilbert transform. Then using Hormander-Mikhlin, the Hilbert transform can be shown to be bounded in Lp . Then the following theorem follows by the transference: Theorem A.1.6 If 1 < p < , We have PN
Lp Lp
<
cp
71
The following inequality is called the Bernstein type inequality, or reverse inequality. Theorem A.1.7 Let 1 < p q < or 1 < p < q , then PN f
q
cp N d(
qp ) pq
f
p
Proof We simply interpolate between A.1.3 and A.1.6. We select p r q, such that 1 < r < then there exists 0 1 such that 1 = , q r and 1 = + 1 . p r or 1 =
1 p
1 . Thus, noting that PN q
Lr Lr
c and PN
L1 L
N n , the conclusion
follows by A.1.2. The Bernstein inequality is just the equivalence of a pair of norms in the nite dimensional vector space with an explicit constant. However, it is also instructive to view this as one manifestation of the uncertainty principle, which states that if a function is localized about the origin in the frequency space, then it must have a large physical support. But note that larger the p, the less sensitive the Lp norm becomes to the size of the physical support. Thus, the Lp norm of the frequency localized functions becomes less sensitive to p as p becomes larger, which is indeed shown by the constant dependence in the Bernsteins inequality. We now turn to a host of inequalities that involve fractional derivatives. We dene the fractional dierentiation operator using multiplication by |k|s in the frequency space: 2 | |s f = |k|s f eikx 2
for each f for which the right-hand-side is in L2 . We note in particular that = | |2 . We note that the fractional dierentiation behaves in the following manner when composed with PN . This is also another type of Bernstein inequality. Theorem A.1.8 Let s 0, then | |s PN f
p
N s cp f
p.
72
Proof Dene j (xj ) = and j is in C . Let = j j . We can express |k|s |k| N = N s (|k|N 1 )s (N 1 k)|k| N . Note that due to the dilation symmetry of the Fourier transform, ((|k|N 1 )s (N 1 k)) (x) = (|k|s (k)) (N x)N n . Thus, it suces to consider the multiplier |k|s (k), which is a symbol of order 0. Hence, by the Hormander-Mikhlin theorem, it is bounded in Lp . Thus, | |s PN
Lp Lp
1 |xj | 1 , 0 |xj | 2
= TN s (|k|N 1 )s (N 1 k) PN PN
Lp Lp
Lp Lp
N s T|k|s (k)
Lp Lp
N s cp .
and the claim follows by the transference principle. On the side note, instead of using the Hormander-Mikhlin theorem, for example in the case s > 1 we can show the bound for which the constant depends on s using a more elementary method. This is because for s > 1, (|k|s (k)) belongs to L1 , so Youngs inequality suces. We thus see that both versions of Bernstein inequality is a way to trade-in the localization in frequency space, for either a gain in integrability or dierentiability. The result that is in a sense opposite to reverse inequalities are the Jackson-type inequalities from the approximation theory. The following theorem is one version of this. Theorem A.1.9 With the same hypothesis as in A.1.8, | |s (I PN )f Proof Let j (xj ) = and j is in C . Let = j j . 73
p
cp N s f
p.
exj 1
2
|xj | 0 |xj | 1
The multiplier can be expressed as |k|s |k| >N = N s (|k|N 1 )s (N 1 k)|k| >N . Again, due to the dilation symmetry, ((|k|N 1 )s (N 1 k)) (x) = (|k|s (k)) (N x)N d . Note that due to the rapid decay of (k) at the origin, |k|s (k) C . But we have |k|s (k)eikx = =( ix d+1 ) |x| |k|s (k)eikx ( ix |x|
d+1 ikx e k)
d+1 s ikx k (|k| (k))e
c(|x|(d+1) ),
for |x| > t and is bounded for |x| < t since |k|s (k) is bounded. Therefore, (|k|s (k)) L1 (Rd ). Consequently, T|k|s (k) | |s (I PN ) N s T|k|s (k)
Lp Lp
c by Youngs inequality. Thus, = TN s (|k|N 1 )s (N 1 k) (I PN ) I PN
Lp Lp Lp Lp
Lp Lp Lp Lp
N s cp .
and the claim follows by the transference principle. Signicance of these two inequalities can be illustrated with one example. We will use these results to prove the Gagliardo-Nirenberg inequality. The more general, Besov version of the inequality is proved by [27, Machihara]. Lemma A.1.10 Let , , p, q, r, satisfy, 1 < q, p r < , 0 < < 1. 1. 0> 2. 0 3. ( Then, f
Lr
<
rq n , rq rp n + , rp
n n n n + ) + (1 )( + ) = 0. p r q r
|f (0)| + | | f
Lq
| | (f P0 f )
1 Lp .
74
Proof We introduce the operator P2k = P2k+1 (I P2k ). Then,
f
r
P0 f +
k=0
P2k f |f (0)| +
k=0
P2 k f r .
Let t > 0 to be chosen later. We split the sum into high-frequency and low-frequency parts and estimate them dierently. First, using the rst condition in our hypothesis, Jackson and Bernstein inequality, P2 k f
klog t
<
r
<
2k(
klog t
rq n) rq
| | f
q
t
rq n rq
| | f q .
Similarly, for the low-frequency part we use the second condition in our hypothesis, P2 k f
0k<log t
<
r
<
2k(
0k<log t
rp n+) rp
| | (f P0 f )
p
t
rp n+ rp
| | (f P0 f ) p .
Let a = ( rq n ), b = ( rp n + ), then we want to minimize rq rp ta | | f
q
+ tb | | f
p,
with respect to t. Optimizing this, we choose t = ( b we get f Choosing =
b a+b r
<
a | | f ) q | | (f P0 f )
q
)1/(a+b) . Plugging this in,
a a+b
|f (0)| + | | f
q
b a+b
| | (f P0 f )
p
.
we see that (a) + (1 )b = 0,
which is the third condition in our hypothesis.
A.2
Nonlinear monotone operator
We set p 2 to be the degree of nonlinearity of our viscosity operator. Also, for convenience, introduce the number q where p + q = pq, i.e. q is the Holder conjugate of p. The nonlinear viscosity operator is dened as follows: (| u|p2 u) = 75 (( u : u)(p2)/2 u)
By (A : B) we mean a componentwise inner product for the matrix A and B. The nonlinear viscosity operator is a type of monotone operator and satises a host of important inequalities that become important in the proof of well-posedness. First, we consider some algebraic inequalities for vectors. See, [8, DiBenedetto] Lemma A.2.1 Let p 2, then for all a, b Rd , there exists > 0 independent of a, b such that 1. (|a|p2 a |b|p2 b, a b) |a b|p 2. (|a|p2 a |b|p2 b, c) (p 1)|c||a b|||a| + |b||p2 Some of the consequences of such a vector inequality are the following: Lemma A.2.2 1. 2. (| u|p2 u | v|p2 v, Proof | u|p2 u | v|p2 v, ( = w dx w) (p 1) w
p
(u v)
p p
(| u|p2 u | v|p2 v,
(u v))
(u v) p (
u
2 p
+
v 2 )p2 p
| w|| (u v)|(| u| + | v|)p2 dx | w|p )1/p ( w
p
| (u v)|p )1/p (
p
|| u| + | v||(p2)p/(p2) )(p2)/p
p2 p
(u v)
| u| + | v|
Another remarkable property of this monotone operator is that it remains a coercive operator when tested against u in the following sense: 76
Lemma A.2.3 Let u (C 2 )n then ( | u|p2 u, u)
i,j,k
| u|p2 (kj ui )2
Proof Note that we have, ( | u|p2 u, u) = (| u|p2 u, u) =
i,j,k
| u|p2 j ui kk j ui dx | u|p2 (kj ui )2
=
i,j,k
+
k
(p 2)| u|p4
i,j,k,l
kl um l um kj ui j ui dx
i,j,k
| u|p2 (kj ui )2 dx
The last line is due to the fact that the sum inside the integral is non-negative. Dene Ip (u) =
i,j,k
| u|p2 (kj ui )2 .
We then have the following embedding theorem: Lemma A.2.4 u Proof | u|p2 (kj ui )2 = |k ui |p2 (kj ui )2 (A.1)
3p
Ip (u)1/p .
1 2 ( j |k ui |p/2 )2 ( (|k ui |p/2 )6 ) 3 , p
where we have used the Sobolev inequality in the last line.
A.3
Compactness and measure theory results
The following is a standard measure theory result [24, Lieb and Loss].
77
Lemma A.3.1 Let ui be a Cauchy sequence in L1 , then there exists u L1 and a subsequence uij such that uij u almost everywhere. Proof The point is to choose a subsequence for which the successive dierence becomes thin very fast. Since ui is Cauchy, we can choose a subsequence uij so that uij+1 uij Clearly,
N L1
2j .
uiN (x) = ui0 (x) +
i=1
uij (x) uij1 (x).
Let F (x) = |ui0 (x)| + uiN (x) dx
|uij (x) uij1 (x)|, then F (x) dx = |ui0 (x)| + |uij (x) uij1 (x)| dx C.
Thus F is integrable, and hence nite almost everywhere. Thus, for almost every x, the sum ui0 (x) + u L1 . Given a sequence of functions ui in L1 (), we call this sequence uniformly integrable if for any > 0 there exists > 0 such that for any M such that |M | < , |
M N i=1
uij uij1 (x) is absolutely convergent. Therefore, for such x there exists
u(x) such that uiN (x) u(x). Since uiN F , dominated convergence theorem implies that
ui dxdt| < ,
for all i. A nice property of the uniformly integrable sequence of functions is that if they converge pointwise, then the integral also converges. Lemma A.3.2 Let ui be uniformly integrable. || < . Suppose there exists u L1 () such that ui u almost everywhere. Then, ui u.
Proof Let > 0 be as in the denition of uniform integrability. By Egoros theorem, there exists a set M such that |M | and ui converges uniformly to u on M c . Now, by Fatous lemma, |u| lim inf
M M
|ui | .
78
now, take i large enough so that |ui u| |M c |1 on M c , then (ui u) +
M Mc
(ui u) 3 .
Since
was arbitrary,the lemma follows.
We also use the Aubin-Lions compactness theorem [28, Malek et al] extensively: Theorem A.3.3 (Aubin-Lions) Let 1 < , < . Let X be a Banach space, and let X0 , X1 be separable and reexive Banach spaces. If X0 X X1 , Then the set {v L (I; X0 ); is compactly embedded in L (I; X). dv L (I; X1 )} dt
A.4
Local averages and Hlder continuity o
We summarize some of the mathematical results that are used in the analysis of the innite Prandtl-number equation. We rst dene the parabolic cylinder Q = {(x, t) : |x x0 | < , |t t0 | < 2 } . First we note the characterization of the Holder norm by the growth of local integrals due to Campanato. Theorem A.4.1 (Campanato) We have the following characterization of the Hlder o semi-norm: [f ]C (Tn ) (sup rnp
r
|f (f )r |p )1/p
A corollary of this theorem is that we have the following characterization of the Hlder o continuity that is more suitable in the parabolic setting. [23, p. 50] Lemma A.4.2 Let 0 < < 1. Suppose we have that for all 0 < r < R. |Qr |1
Qr
|u (u)QR |2 Cr2
Then, for all 0 < r < cR. |u(x, t) u(y, s)| C2 (|x y| + |t s|/2 ) Thus, u is -Hlder continuous in space and /2-Hlder continuous in time. We let the o o smallest constant on the right hand side the C ,/2 semi-norm of u. 79
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BIOGRAPHICAL SKETCH
Yuki Saka Yuki Saka was born on June 9th 1980 to Masakatsu and Yumiko Saka, in Nagano prefecture Japan. He received his Bachelors degree in computer science and pure mathematics from the University of California, Berkeley in spring 2003, Masters in applied mathematics from the Florida State University in spring 2007. His research interests include mathematical and numerical analysis of partial dierential equations, applied harmonic analysis and scientic computing.
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