Feb 8, 2000
A Universal Law for the Tails of the PDFs in Multidimensional Burgers' Turbulence
Uriel Frisch, (Obs. Cote d'Azur-Nice)
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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.
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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.
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Coauthors on this work: J. Bec, K. Khanin, B. Villone
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cond-mat/9912110, astro-ph/9910001, cond-mat/9906047
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