Linear and Quasilinear Parabolic Problems

In this treatise we present the semigroup approach to quasilinear evolution equa of parabolic type that has been developed over the last ten years, approxi tions mately. It emphasizes the dynamic viewpoint and is sufficiently general and flexible to encompass a great variety of concrete systems of partial differential equations occurring in science, some of those being of rather 'nonstandard' type. In partic ular, to date it is the only general method that applies to noncoercive systems. Although we are interested in nonlinear problems, our method is based on the theory of linear holomorphic semigroups. This distinguishes it from the theory of nonlinear contraction semigroups whose basis is a nonlinear version of the Hille Yosida theorem: the Crandall-Liggett theorem. The latter theory is well-known and well-documented in the literature. Even though it is a powerful technique having found many applications, it is limited in its scope by the fact that, in concrete applications, it is closely tied to the maximum principle. Thus the theory of nonlinear contraction semigroups does not apply to systems, in general, since they do not allow for a maximum principle. For these reasons we do not include that theory.

Contenu
Notations and Conventions.- 1 Topological Spaces.- 2 Locally Convex Spaces.- 3 Complexifications.- 4 Unbounded Linear Operators.- 5 General Conventions.- I Generators and Interpolation.- 1 Generators of Analytic Semigroups.- 2 Interpolation Functors.- II Cauchy Problems and Evolution Operators.- 1 Linear Cauchy Problems.- 2 Parabolic Evolution Operators.- 3 Linear Volterra Integral Equations.- 4 Existence of Evolution Operators.- 5 Stability Estimates.- 6 Invariance and Positivity.- III Maximal Regularity.- 1 General Principles.- 2 Maximal Hölder Regularity.- 3 Maximal Continuous Regularity.- 4 Maximal Sobolev Regularity.- IV Variable Domains.- 1 Higher Regularity.- 2 Constant Interpolation Spaces.- 3 Maximal Regularity.- V Scales of Banach Spaces.- 1 Banach Scales.- 2 Evolution Equations in Banach Scales.- List of Symbols.