On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Kenneth Hvistendahl Karlsen,Nils Henrik Risebro

doi:10.3934/dcds.2003.9.1081

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro

Open Access

https://doi.org/10.3934/dcds.2003.9.1081

Copy DOI

Journal: Discrete & Continuous Dynamical Systems - A	Publication Date: Jan 1, 2003
Citations: 164	License type: cc-by

Abstract

We study nonlinear degenerate parabolic equations where the flux function $f(x,t,u)$ does not depend Lipschitz continuously on the spatial location $x$. By properly adapting the 'doubling of variables' device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form $k(x)f(u)$, where $k(x)$ is a vector-valued function and $f(u)$ is a scalar function.

Highlights

The main subject of this paper is uniqueness and stability properties of entropy solutions of nonlinear degenerate parabolic equations where the flux function depends explicitly on the spatial location
The problems that we study are initial value problems of the form ut + divf (x, t, u) = ∆A(u) + q(x, t, u), (x, t) ∈ ΠT = Rd × (0, T ), u(x, 0) = u0(x), x ∈ Rd, where T > 0 is fixed, u(x, t) is the scalar unknown function that is sought, f = f (x, t, u) is called the flux function, A = A(u) is the diffusion function, and q = (x, t, u) is the source term
In the present paper we generalize Carrillo’s uniqueness result [12] by showing that it holds for the Cauchy problem with a flux function f = f (x, t, u) where the spatial dependence is nonsmooth

Summary

Introduction

The main subject of this paper is uniqueness and stability properties of entropy solutions of nonlinear degenerate parabolic equations where the flux function depends explicitly on the spatial location. An important step forward in the general case of A(·) being merely nondecreasing was made recently by Carrillo [12], who showed uniqueness of the entropy solution for a particular boundary value problem with the boundary condition “A(u) = 0”. In the present paper we generalize Carrillo’s uniqueness result [12] by showing that it holds for the Cauchy problem with a flux function f = f (x, t, u) where the spatial dependence is nonsmooth (non-Lipschitz).

Results

Conclusion