Burgers system driven by a periodic stochastic flow

Abstract

Burgers System (BS) is Navier-Stokes system without pressure and in-compressibility. It is one of the most popular models of hydrodynamics and has a lot of applications. In this paper, we consider BS driven by a potential force whose potential is a periodic stochastic flow. If x = (x}, ... xn) is the vector of coordinates and U = (Ub • • • , un) is the velocity vector then the n-dimensional BS takes the form au . at + (u,V)u = p.Llu + VB(x,t) (1) or in the coordinate form n aUi L aUi a . -+ -. Uk = P.LlUi + -a B(x,t). at k=l aXk Xi (2) Here and later V means gradient with respect to space variables, dot means the differentiation with respect to time, a is the differential with respect to time variable, p. is the viscosity. We impose periodic boundary conditions with period 1 for each Xi. The potential B(x, t) is the generalized derivative of a periodic stochastic flow. The flow itself can be represented in the form of Ito stochastic differential aB(x, t) = L Ck e 21ri (z,k) dBk(t) (3) kEZ n In the last expression Bk(t) are standard complex Wiener processes, B-k(t) = Bk(t) and independent otherwise. About the coefficients Ck we assume that Co = 0, C-k = Ck and Ck decay so fast that the moment (4) These conditions imply in particular that B(x, t) is a.e. a C2-smooth real-valued periodic function of x. When physicists write about BS with a poten-tial force which is "white noise" in time they actually keep in mind forces generated by potentials which we just described. An advantage (or disadvantage) ofBS is due to the fact that it leaves invariant the space of gradient solutions. In the non-random case this means that if initially u(X; 0) is gradient of some function then u(X; t) is also gradient for N. Ikeda et al. (eds.), Itô's Stochastic Calculus and Probability Theory