Solutions for quasilinear nonsmooth evolution systems in \(L^p\)
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.
three-dimensional polyhedral space domain
scientific article; zbMATH DE number 2102294