Regularizing Effects for u sub t + A(psi(u))=0 in L1 Vector Space.
Abstract
Various initial-boundary value problems and Cauchy problems can be written in the form du/dt + A(psi (u))=0 where psi=R approaches R is nondecreasing and A is the linear generator to strongly continuous nonexpansive semigroup e to the -ta power in a L1 vector space. Many models of interesting phenomena yield equations for the evolution of a system of this abstract form where psi is a nonlinear nondecreasing function and A is an operator. E.g., A may be the Laplacian (perhaps under boundary conditions) or A may be del/del x, while psi may be a power law. Models like this occur in porous flow, plasmas and conservation laws. In this work it is shown that a broad class of such problems are solvable by the nonlinear semigroup theory. The main point, however, is a regularizing effect which estimates the speed of the system at time t > 0 by the integral of the initial data. This has consequences for the regularity of the solutions of concrete problems and their asymptotic behaviour.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1981
- Accession Number
- ADA099349
Entities
People
- Michael G. Crandall
- Michel Pierre
Organizations
- University of Wisconsin–Madison