The Inviscid Burgers Equation with Initial Value of Poissonian Type

Abstract

We study the discontinuities (shocks) of the solution to the Burgers equation in the limit of vanishing viscosity (the inviscid limit) when the initial value is the opposite of the standard Poisson process p. We show that this solution is only defined for t e (0, 1). Let T 0 = 0 and T n , n≧1, be the successive jumps of p. We prove that for all M > 0 the inviscid limit is characterized on the region x e (-∞, M], t e (0, 1) by the increasing process $$N(t) = \sup \{ n \in \mathbb{N} {\text{| }}M + nt > T_n \} $$ and the random set I(x) = {n e {0,..., N(t)}‖T n -nt≦x<T n+1 - nt}. The positions of shocks are given in a precise manner. We give the distribution of N(t) and also the distribution of its first jump. We also prove similar results when the initial value is u μ(y, 0) = -μp(y/μ2) + μ-1 max(y, 0), μ e (0, 1).