Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations

Abstract

Consider the equation: ut=νΔu-(u·∇)u+f for x∈[0,1]n and t∈(0,∞), together with periodic boundary conditions and initial condition u(t, 0) = g(x). This corresponds a Navier-Stokes problem where the incompressibility condition has been dropped. The major difficulty in existence proofs for this simplified problem is the unbounded advection term, (u · ∇)u. We present a proof of local-in-time existence of a smooth solution based on a discretization by a suitable Euler scheme. It will be shown that this solution exists in an interval [0, T), where T ≤ 1/C, with C depending only on n and the values of the Lipschitz constants of f and u at time 0. The argument given is based directly on local estimates of the solutions of the discretized problem. © 2007 Springer-Verlag Wien.