Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 147, pp. 1-32.
Title: Renormalized entropy solutions for degenerate nonlinear
evolution problems
Author: Kaouther Ammar (Technische Univ. Berlin, Germany)
Abstract:
We study the degenerate differential equation
$$
b(v)_t -\hbox{ div}a(v,\nabla g(v))=f \quad
\hbox{on }Q:= (0,T) \times \Omega
$$
with the initial condition $b(v(0,\cdot))=b(v_0)$ on $\Omega$ and
boundary condition $v=u$ on some part of the boundary
$\Sigma:=(0,T) \times \partial \Omega$ with $g(u)\equiv 0$ a.e. on
$\Sigma$. The vector field $a$ is
assumed to satisfy the Leray-Lions conditions, and the
functions $b,g$ to be continuous, locally Lipschitz, nondecreasing
and to satisfy the normalization condition $b(0)=g(0)=0$ and the
range condition $R(b+g)=\mathbb{R}$.
We assume also that $g$ has a flat region $[A_1,A_2]$
with $A_1\leq 0\leq A_2$. Using Kruzhkov's method of doubling
variables, we prove an existence and comparison result
for renormalized entropy solutions.
Submitted August 15, 2009. Published November 20, 2009.
Math Subject Classifications: 35K55, 35J65, 35J70, 35B30.
Key Words: Renormalized; degenerate; diffusion;
homogenous boundary conditions; continuous flux.