W09_201B_HW5sol thomases

W09_201B_HW5sol thomases - P0814. HOME WOrk 5 Sofufions...

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Unformatted text preview: P0814. HOME WOrk 5 Sofufions jaflpo AWJ MAT 20! B A Becca T/vomases “ Winter 2007 ,gume mg A 00113. Jiawei Z/wang r Hem I P O 13.93 LU: M, Nbe c/LZeJ Sodast 9% a HIT/9M Since WM a we Orflojonal projections wifiw ran p r/M , ran (2 2 M. Pron that/f HR fol‘wj Cow'b’ifions are afloat/DEM ~. ((1) M C IV]. (5) QP: p; (C) p0 :P; W (W (of) UPXU $ [IQXl‘fvr aLL 7< (— H, (e) (y, PX) XGH. Proof. U (a) :5) up)»; WQ WM Me, 442 fad {l‘eci’uavflfl ; ’X owl 0:19 ,‘f x :Px’ {of my PmJ‘M‘on Ver’ HzranPEBkerP SI X,6WP=M; m1 Xlekerp 9+. X —__-— «x, + “x1 , Hence, we have S (x, @x) for all / QP’X : ad)“ 1&(P(X’fl(l)) 2&(PXI+PX1)2&(XI+O) : 02X». Stme so (EA/(CM:er I we have 62X! : X‘ . ‘ It” fanwS ‘wat ‘apx: any: mam? = PM 0 = Px.+ PK; : POQ‘sz) :PX. TlW/Q‘EWQI &P= P- / “(D—.7 La)"; From (20, we have 62th @x‘ ’ QM QP:P, anal thch.+xz)=Px,=y,/ we hm 9’8ng (>2 X‘ 1’. X. 7% and X; 6 amp. Tim (‘5 etiw‘vNem‘ +0 H X. 6 WP -‘> X: 6 WW”. Heme, M C‘ M. 5/ "(09 => (0‘5. V 'xg H, H = WP (—9 k9,?) whoa/q X}: X¥)+ XP) Mere X;}”6von&, x? 6 ma Gmeél @kQ/(fa j IE fol/LOWS HUN" » Fax: Paémxn = PCQXMQxl) ‘3 P(X( ‘f‘ QCX§)+X(§)3) : PO<I+ X£)+ O) : sz‘f’ Pxi” = Pmo v : PXI+ PXI 7* PX. S‘Mm 7C is arbh‘mv), we Ma Pa = P, ' \/ (ICC) t) (0’)" gm #‘tm pazP, we haw Fr 0015 X6”, P62 X = PX, the norm on bow snake, WQ 010M“ (I PQXU = HPYH. SIM; UPch we have UPQQX)“ é (IQXH , Heme “qu = 1mm)“ “é um“) V. {la 0M 70 e34. r9036 “(o?)=>(€)"., HPxH é (HM! fwaM «EH, ll?)le s “()2qu 12x au X614, No‘ve W, HPXHII <Px/ Px> = < X, Pix) =09 Px7, “(so we [we “emf: <x, am, hem <X,Px> $ (39050 fora“ XGH. "(6 a £61)" .. 999m M czf N , +hw W em a ‘X M W ’X &.N, Left 7C: ’Xl + "XL, where we Vanfl , x1 e/cgro'li (x, (2x7 2 <x, elm—rm» 7- 09 x.) + (x, 0> '1 (X, Y.) < (V, X) 91mm <y, x17 7 o~ ( <xn<n= mm =<x, Q‘xv “NY/<90 7 O). and (Y,X7=(¥,Xa>+()®(27. Tim‘s is ammo" +9 +Le 7am HM— <X/X7 : (x, PX) S (X, (XX) W 4 Ex 9.4. ProUem ‘2 §Q1>P0§e Hm ( Pu) 1‘s CK 3901qu of orflmgmd Py/OJEQfionS OH 6: HJLM Spica H gm HM m le?nfl :3 Wm Pm U VthPn = )“P, ’ ’11:! Pave W ( Pm) WWW/a Sfmlylj 4’0 44w Mammy operW/w I A4 V19 00, £5“ +wa 0(«09/3 IM Com/9er +0 1418 (delay 0PM Wfik “9ij +0 #9 norm Wham Pr) = I 7% M" £ufi0da-H9 (£er I’l. Plow}. {801' 0( élnf] Le er ECUIS 91. YUmPn GaulA n 6 N/ mch chzcmcthm ’ \7/ ’X EH, we have /X t: 2 ad Q4 ole: ' Hence , W X" = nguzwau _. 2 “Elm 0“ H : I “1941 JERIn W 3 W": 0. W °b elem“ ,v‘ [\me v Y“ -‘2 0 Cu) IA») 00. “Wake; (Pu) Combjm 5mm +49 1 _,-r P6813 SW04: VM 7°» '3 m7/J 5+ PA :1) WLMA mpg») '11 m e keer m xnio‘ Havth / H Pm Xn — M! ‘ H 0 4w fix Hm} " m— :1. (7?) We can oLw‘mm 01 S€€qem2 (Xq B “F > Suf- /\$ SAfiSf'Qofl 74w 9&64 KL L3 “ (M am) I H Pvlll Q : My 1M1” {79%er [W Z W” - ((anl v \ PM dose wo'f (bu/(3:692 4"? I wH—L YQ/QPe/cl” 10 “HQ O’HWLN mm W Exercise 8.5 Let ’H = L2(T3;R3) be the Hilbert space of 21r-periodic, square- integrable, vector—valued functions u : T3 -—+ R3, with the inner product (u, v) = / u(x) - v(x) dx. T3 We define subspaces V and W of ’H by V = {veC°°(T3;Ra)|V-v=0}, W = {WEC°°(T3;R3)|w=V<pforsorne<pzT3—-+R}. Show that ’H = M @N’ is the orthogonal direct sum of M = V- and N = W. Let P be the orthogonal projection onto M. The velocity v(x, t) e R3 and pressure p(x,t) e R of an incompressible, viscous fluid satisfy the Notifier-Stokes equations Vc+v~Vv+Vp=uAv, V-v==0. Show that the velocity v satisfies the nonlocal equation v; + P[v - Vv] = VAv. We first show that V and W are orthogonal. Let v E V and w E W. Then, integrating by parts, the inner product (v,w)= f13u(x)-v(x)dx = (p(x)v(x)|T3 .. fT3(V.v)V¢dx = 0, since V is 2m periodic (so the surface term vanishes) and V - v = 0 (so the integral also vanishes). Since the kernel of an orthogonal projection closed, we have N = W C ker PV and M = \7 C ker PW. Next, we show that W J. V. To see this note that for any limit point v’ of V there exists a sequence {vn} E V such that vn —> v', and v' is orthogonal to W since (v’,w) =<lir_gv,,,w> = lim(vn,w) = 0 for all w E W. Futhermore W J. V since for any limit point w’ of W there exists a sequence {wn} E W such that wn ——> v', and w' is orthogonal to Vsince (w’,v) = (lim wn,v> = lim(wn,v) = 0 for all v E V. Now ifx E W H V—Lthen since x E W, xJ. l7, and sincex E \7, xi W, so it must hold thatx=0, thus W O V = {0}. F036 9' It remains to be shown that W U \7 spans H. Since H = L2(T3, R3), it follows that has a Fourier basis {exp[in.-(x+y+z)]i + exp[inj(x+y+z)]j + exp[ink(x+y+z)]k}. So expanding about this basis, we see that if v E V, then v = Eunieim'm’mi + E anje""'(x+y+2)}t + Eameimmymk and then "k n, "h "1' I _ . in (x+y+z)-.‘ . in-(x+y+z)'t . ' (x+ + )d . V v—Emianie ’ 1+Eznjanje ’ 1+2mkankem‘ y zk. This can only be n. I zero if a,” = ani = a,“ = 0, when ni,nj,nk are nonzero. So v =anii + a,” j + ankk , where the coefficients are any real number. On the other hand if w E W, then _ I _', . '( )1 r a w = Eame'w‘+y ")1 + Zanje'" “y H _] + Zane'mmymk =V(p for some real-valued ,,. "t J "" function (p. But this implies that f Eanlein,(x+y+z)? + EanjeinJ-(nyu); + Bankein.(x+y+z)k dx = fTS dexe T3 :1; nj nk ' eiru: But fem‘dx = —.— , so the left hand side of the above equation only gives a real value if T T a =anj =anh =0 whenni=nj=nk=0. "I m Therefore any vector x E H can be expressed as x = v + w, where the v E Vpart contains the Fourier components for ni = nj = nk = 0, and that w E W part contains the Fourier components when ni,nj,nk are nonzero. So we can conclude that H =W (43 V = M @N. Given the equations v, + v-Vv +Vp = vAv and V - v: 0, we project rewrite the first equation as vt + v- Vv +Vp - vAv = 0 so that P(vt + v - Vv + Vp - vAv) = 0. Now the time derivative commutes with the divergence, so vt E V and P(vt)= vt. Since the pressure p is a scalar function, we have Vp E W which implies that P(Vp) = 0. Next, V-vAv=(V-v)vAv+(VvA)- v= (V-v)vAv+vA(V-v)=0, so V-vAv e v and P(V - vAv) = V - vAv. So by the above relations, we have P(vt + v- Vv + Vp — vAv) = vt + P(v - Vv) —- vAv = 0, and we can conclude that vt + P(v - Vv + Vp) = vAv. Exercise 8.9 Let A C H be such that M = {2: E H | a: is a finite linear combination of elements in A} is a dense linear subspace of H. Prove that any bounded linear functional on H is uniquely determined by its values on A. If {no} is an orthonormal basis, find a. necessary and sufficient condition on a family of complex numbers ca for there to be a bounded linear functional <p such that (Mun) = ca. ch that cp(x) = cp’(x) for all x E A. Now let y E AC. Then since M is dense in H there exists a sequence {xn} E M such that xn ->y. Then since both (p and cp’ are linear we have W) = wqig} x.) = gig; w.) = gig;an akz.» = gig: we...) k k = gig; We...) = <p’qi3; x.) = My) 1: where Eakznk is a finite linear combination for xn k E M. Thus, we see that q)(y) = CP'O’) for all y E AC , showing that cp is uniquely determined by its values on A. Since the span of an orthonormal basis for H is dense in H, it follows that any bounded linear function is uniquely determined by its values on the basis. Let (p(x) = <Ec u x>. (111’ Then it is clear that <p(ua) = ca. Since {ua} is a basis for H, there exists y = Ecaua E H and (p(y)=<2caua,2caua> = Elcalz. In order for Elcml2 to converge, only a (1 em. But in order for cp(x) to be bounded for every x = Zbaua, it follows that only a finite number of cacan be nonzero. The reason is because for every set {ca} containing infinitely (but countable) nonzero terms such that series Elca] < 00, we can an x = Ebaua, where bu = l/c—a and then zc—aba does not a a converge. Therefore in order for to be bounded it is necessary for {Cu} to have only finitely many nonzero terms. On the other hand if {ca} does have only finitely many nonzero terms then llinrnlrp(x)|=llilr_r11<2cmum,x>=c=c”x“ where c = max{ca}. So it (1 follows that cp is bounded. So a necessary and sufficient condition for there to be a bounded linear operator cp such that <p(ua) = Cu is that the family of complex numbers ca} has only finitely many nonzero terms. Exercise 8.11 Prove that if A : H -—r H is a linear map and dim’H < 00, then dimkerA+dimranA = dim'H. Prove that, if dim'H < 00, then dimkerA = dim ker A‘. In particular, kerA = {0} if and only if ker/1“ == If dim H < 00, then there exists some n E N such that dim H = n, so then the linear map A can be expressed as an n x 11 matrix. But then the dim ran A = rank of A and dim ker A = nullity of A. So by the rank-nullity theorem, dim ker A ' ' detail, since dim H < 00, then is finite and A is a bounded ' W 9 Theorem 8.17, ran A = (ker A*)*. Furthermore H = ker A* 69 ran A. Now dim ran A = dim ran A and dim ker A* = dim ker A (as shown below) so dim H = dim ker A + dim ran A. Since the kernel of a matrix is invariant under transposition and solutions to linear equations are invariant under complex conjugation, it follows that the nullity of a A is equal to the nullity of A*, when dim H < 00. Thus, dim ker A = dim ker A“. In particular, dim ker A = 0 iff ker A = {0}, and likewise for A*, so it follows that ker A = {O} iffker A* = {0}. Exercise 8.12 Suppose that A : H —+ H is a bounded, self-adioint linear operator such that there is a constant c > 0 with alkali 5 “A2” for all a: E H. Prove that there is a unique solution a: of the equation Ax = y for every y E H. By Proposition 5.30, A has a closed range and the only solution to the equation Axu= 0 is x = 0. Since A is self-adjoint, a similar statemfltiiolds for A*. So it follows that kerA* = {0}. It follows from Theorem 8.17 that ran A = (ker A*)1 = {0}* = H . Now, by Theorem 8.18, since A has a closed range, the equation Ax = y has a solution if y is orthogonal to ker A*. But ker A“ = {0} so every y E H is orthogonal to ker A", and thus for every y E H, x is a solution to Ax = y. Now we show uniqueness. Since A is self-adjoint, it follows that ker A = ker A* = {0}" ' ~ Therefore the operator is one-to-one, and thus every solution is unique. _ “N... “Mum. __..-..~_ ,, _ .“mu—.Wwwemwuuwmmt'“‘“‘““‘““"‘W " “5 M ages- _ , _,,.-,._-_W-mm (at MLE Mal/.gtflm filer--. ~« :1: more m‘ “emanatefiaa a: ” ____ ,, Gé,amaf%zm e. i ,, Ag; “Wamdwflgérw ~ ‘ ‘ Mghzggze-EO- LE)“ Hinge-3 ., é. 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This note was uploaded on 10/12/2010 for the course MATH 201B taught by Professor Thomases during the Spring '10 term at UC Merced.

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W09_201B_HW5sol thomases - P0814. HOME WOrk 5 Sofufions...

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