Electron. J. Diff. Equ., Vol. 2009(2009), No. 147, pp. 1-32.

Renormalized entropy solutions for degenerate nonlinear evolution problems

Kaouther Ammar

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.

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Kaouther Ammar
TU Berlin, Institut für Mathematik, MA 6-3
Strasse des 17. Juni 136, 10623 Berlin, Germany
email: ammar@math.tu-berlin.de, Fax: +4931421110, Tel: +4931429306

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