arXiv:nlin/0209059v3[nlin.CD]28 Jun 2003J. Stat. Phys.in press. (updated version)Dedicated to the memory of Richard Pelz.Singularities of Euler flow? Not out of the blue!U. Frischa,b, T. Matsumotoa,b,cand J. Beca,baCNRS UMR 6529, Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, FrancebCNLS - Theoretical Division, LANL, Los Alamos, NM 87545, USAcDep.Physics, Kyoto University, Kitashirakawa Oiwakecho Sakyo-ku, Kyoto 606-8502, Japan(Dated: October 25, 2018)Does three-dimensional incompressible Euler flow with smooth initial conditions develop a sin-gularity with infinite vorticity after a finite time? This blowup problem is still open. After brieflyreviewing what is known and pointing out some of the difficulties, we propose to tackle this is-sue for the class of flows having analytic initial data for which hypothetical real singularities arepreceded by singularities at complex locations.We present some results concerning the natureof complex space singularities in two dimensions and propose a new strategy for the numeri-cal investigation of blowup.(A version of the paper with higher-quality figures is available at)I. PHENOMENOLOGY OF BLOWUPAccording to Richardson’s ideas on high Reynoldsnumber three-dimensional turbulence, energy introducedat the scaleℓ0, cascades down to a scaleη≪ℓ0where it isdissipated. Consider the total timeT⋆which is the sum ofthe eddy turnover times associated with all the interme-diate steps of the cascade. From standard phenomenol-ogy `a la Kolmogorov 1941 (K41), the eddy turnover timevaries asℓ2/3. If we let the viscosityν, and thusη, tendto zero,T⋆is the sum of an infiniteconvergentgeometricseries. Thus it takes afinite timefor energy to cascadeto infinitesimal scales, an observation first made by On-sager [1].We also know that in the limitν→0, theenstrophy, the mean square vorticity, goes to infinity asν−1(to ensure a finite energy dissipation).From such observations, it is tempting to conjecturethat ideal flow, the solution of the (incompressible) 3-DEuler equation∂tv+v· ∇v=−∇p,(1)∇ ·v= 0,(2)when initially regular, will spontaneously develop a sin-gularity in a finite time.This is of course incorrect: the kind of phenomenol-ogy assumed above is meant only to describe the (sta-tistically) steady state in which energy input and en-ergy dissipation balance each other; the inviscid (ν= 0)initial-value problem is not within its scope.Anotherpossible argument in favor of singularities has to do withthe scaling properties of the high Reynolds number so-lutions (e.g. thek−5/3spectrum).For the simpler caseof the Burgers turbulence [2], the scaling of spectra andstructure functions is clearly rooted in the singularities(shocks) appearing in the solutions in the limit of van-ishing viscosity. It is however well-known that power-lawbehavior can be present without any singularities.Anexample is the Holtsmark process, that is any compo-nent of the electric or gravitational field produced at agiven point by a set of charges or masses with an initialPoisson distribution in space and moving with uniformindependent isotropic velocities (having, e.g., a Gaussiandistribution). It is then easily shown (by adaptation ofthe technique used by Chandrasekhar [3]) that the cor-relation function is∝ |t−t′|−1. This power-law behaviorcomes from the algebraic distribution of the distances ofclosest approach to the point of measurement and not

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