Renormalized entropy solutions for degenerate nonlinear evolution problems

Ammar, Kaouther

Electronic Journal of Differential Equations, Tome 2009 (2009).

Résumé

Summary: 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.

Détail

Classification : 35K55, 35J65, 35J70, 35B30
Keywords: renormalized, degenerate, diffusion, homogenous boundary conditions, continuous flux