Regularizing Effects for u sub t + A(psi(u))=0 in L1 Vector Space.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Various initial-boundary value problems and Cauchy problems can be written in the form dudt Apsi u0 where psiR 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 deldel 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.
- Theoretical Mathematics
- Fluid Mechanics