PDFs for the Random Forced Burgers Equation

We propose a new approach for the analysis of stationary correlation functions of 1D Burgers equation driven by a random force. We use this to study the asymptotic behavior of the probability distribution of velocity gradients and velocity increments. PACS number 47.27.Gs, 03.40.Gc Statistical properties of solutions of random forced Burgers equation have been a subject of intensive studies recently (see [1, 2, 3, 4, 5, 6]). Of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments. Aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces [1], it is also hoped that the eld-theoretic techniques developed for the Burgers equation will eventually be useful for understanding more complex phenomena such as turbulence. In this paper, we propose a new and direct approach to analyze the scaling properties of the various distribution functions for the random forced Burgers equation. We will consider the problem (1) ut + 1 2 (u)x = uxx Vx(x; t) Most of our discussion will be limited to the inviscid case when = 0. But we will summarize at the end the necessary changes for the case when 0 < << 1. The potential V of the force Courant Institute of Mathematical Sciences, New York University. Heriot-Watt University, Edinburgh and Landau Institute, Moscow. Princeton University, Princeton. Princeton University, Princeton and Landau Institute, Moscow.