Solutions for quasilinear nonsmooth evolution systems in \(L^p\)
type: Item
instance of: scholarly article
title: Solutions for quasilinear nonsmooth evolution systems in \(L^p\)
zbMATH Open document ID: 1053.35058
author: Joachim Rehberg
author: Gunther Schmidt
author: V. Maz'ya
author: Johannes Elschner
published in: Archive for Rational Mechanics and Analysis
publication date: 2004-09-22T00:00:00Z
review text: The authors consider nonsmooth quasilinear parabolic systems of the form \[ u_k'-\nabla\cdot(\mu_k I_k({\mathbf u})\nabla u_k)= R_k({\mathbf u},\nabla{\mathbf u}),\quad {\mathbf u}(T_0)={\mathbf u}_0,\quad k=1,\ldots,m \quad\text{in}\;\Omega\times(T_0,T) \] with \(\Omega\) being a bounded three-dimensional polyhedral space domain, and imposing Dirichlet boundary conditions on \(\partial \Omega\) which may depend suitably on time. They establish local solvability in \(C([T_0,T], L^p(\Omega;{\mathbb R}^m))\cap C^1((T_0,T], L^p(\Omega;{\mathbb R}^m))\) under suitable conditions on the piecewise constant coefficients \(\mu_k.\) The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for anisotropic materials. These results are based on sharp pointwise estimates for Green's function, which are also of independent interest. To treat the nonlinear problem, the authors apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As application the authors consider primarily reaction-diffusion systems. The treatment of such equations in an \(L^p\) context seems to be new and allows (by Gauss' theorem) the proper definition of the normal component of currents across the boundary.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 2102294
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: local solvability
zbMATH Keywords: Green's function
zbMATH Keywords: pointwise estimate
zbMATH Keywords: three-dimensional polyhedral space domain
zbMATH Keywords: piecewise constant coefficients
zbMATH Keywords: interface problems
zbMATH Keywords: isotropic materials
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: s00205-003-0297-y
OpenAlex ID: W2158999556
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DOI: 10.1007/S00205-003-0297-Y
description: scientific article; zbMATH DE number 2102294