Weak convergence and deterministic approach to turbulent diffusion. (English) Zbl 0967.35112

Bardos, Claude; Ghidaglia, Jean-Michel; Kamvissis, Spyridon

Guo, Yan (ed.), Nonlinear wave equations. A conference on honor of Walter A. Strauss on the occasion of his sixtieth birthday, Brown University, Providence, RI, USA, May 2-3, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 263, 1-15 (2000).

Summary: The purpose of this contribution is to show that some of the basic ideas of turbulence can be addressed in a deterministic setting instead of introducing random realizations of the fluid. Weak limits of oscillating sequences of solutions are considered and along the same line the Wigner transform replaces the Kolmogorov definition of the spectra of turbulence. One of the main issue is to show that, at least in some cases, this weak limit is the solution of an equation with an extra diffusion (the name turbulent diffusion appears naturally). In particular, for a weak limit of solutions of the incompressible Euler equation (which is time reversible) such process would lead to the appearance for irreversibility. In the absence of proofs, following a program initiated by P. Lax, the diffusive property of the limit is analyzed, on the zero dispersion limit of the Korteweg-de Vries equation and of the nonlinear Schrödinger equation.
For the entire collection see [Zbl 0947.00025].

MSC:

35Q35	PDEs in connection with fluid mechanics
76F20	Dynamical systems approach to turbulence

Keywords:

Wigner transform; incompressible Euler equation; zero dispersion limit; Korteweg-de Vries equation; nonlinear Schrödinger equation