This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W01,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Applied Mathematics