×

Renormalized entropy solutions for degenerate nonlinear evolution problems. (English) Zbl 1182.35147

Summary: We study the degenerate differential equation
\[ b(v)_t -\operatorname{div} a(v,\nabla g(v))=f \quad \text{on }Q:= (0,T) \times \Omega \]
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.

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B51 Comparison principles in context of PDEs
PDFBibTeX XMLCite
Full Text: EuDML Link