Schedule Feb 8, 2000
A Universal Law for the Tails of the PDFs in Multidimensional Burgers' Turbulence
Uriel Frisch, (Obs. Cote d'Azur-Nice)
We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two ($d\le 2$) or higher ($d\ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute universal power-law tails to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. ............................................... At the end of the seminar we briefly discuss the relation between singularities and power-law pdf's in a more general context. For example the possibility to look for singularities in solutions to the Navier-Stokes equation (in the inviscid limit) by examining pdf's of space or time derivatives obtained from experimental or numerical data. Singularities should always give power-law tails (at least as intermediate asymptotics when the viscosity is small). The converse is however not true. ..................................................................................................................... Coauthors on this work: J. Bec, K. Khanin, B. Villone
cond-mat/9912110, astro-ph/9910001, cond-mat/9906047

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