An Oleinik-type estimate for a convection-diffusion equation and convergence to N-waves

Abstract

In this article we propose an Oleinik-type estimate for sign-changing solutions to a convection-diffusion equation u(t) + (\u\(gamma)/gamma)(x) = muu(xx), u(x,0) = u(0)(x), u,xepsilonR, 1<gammaless than or equal to2, u,t>0. Since the Oleinik entropy inequality holds for nonnegative solutions or inviscid case (mu = 0) only, the theoretical progress for the case was limited. In this paper we show that its solution satisfies an Oleinik-type estimate, t(2/gamma)u(x)less than or equal toC, 1<gammaless than or equal to2, t>0, where C = C(u(0), gamma) > 0. Using this estimate, the convergence to an N-wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled. (C) 2003 Elsevier Inc. All rights reserved.