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.- 1.1 Properties of Linear Operators.- 1.2 The Class H(E1E0).- 1.3 Perturbation Theorems.- 1.4 Spectral Estimates.- 1.5 Compact Perturbations.- 1.6 Matrix Generators.- 2 Interpolation Functors.- 2.1 Definitions.- 2.2 Interpolation Inequalities.- 2.3 Retractions.- 2.4 Standard Interpolation Functors.- 2.5 Continuous Injections.- 2.6 Duality Properties.- 2.7 Compactness.- 2.8 Reiteration Theorems.- 2.9 Fractional Powers and Interpolation.- 2.10 Semigroups and Interpolation.- 2.11 Admissible Interpolation Functors.- II Cauchy Problems and Evolution Operators.- 1 Linear Cauchy Problems.- 1.1 Hölder Spaces.- 1.2 Existence and Regularity Theorems.- 2 Parabolic Evolution Operators.- 2.1 Basic Properties.- 2.2 Determining Integral Equations.- 3 Linear Volterra Integral Equations.- 3.1 Weakly Singular Kernels.- 3.2 Resolvent Kernels.- 3.3 Singular Gronwall Inequalities.- 4 Existence of Evolution Operators.- 4.1 A Class of Parameter Integrals.- 4.2 Semigroup Estimates.- 4.3 Construction of Evolution Operators.- 4.4 The Main Result.- 4.5 Solvability of the Cauchy Problem.- 5 Stability Estimates.- 5.1 Estimates for Evolution Operators.- 5.2 Continuity Properties of Mild Solutions.- 5.3 Holder Estimates.- 5.4 Boundedness of Mild Solutions.- 6 Invariance and Positivity.- 6.1 Yosida Approximations.- 6.2 Approximations of Evolution Operators.- 6.3 Invariance.- 6.4 Orderings and Positivity.- III Maximal Regularity.- 1 General Principles.- 1.1 Sobolev Spaces.- 1.2 Absolutely Continuous Functions.- 1.3 Generalized Solutions.- 1.4 Trace Spaces.- 1.5 Pairs of Maximal Regularity.- 1.6 Stability.- 2 Maximal Hölder Regularity.- 2.1 Singular Holder Spaces.- 2.2 Semigroup Estimates.- 2.3 Trace Spaces.- 2.4 Estimates for KA.- 2.5 Maximal Regularity.- 2.6 Nonautonomous Problems.- 3 Maximal Continuous Regularity.- 3.1 Necessary Conditions.- 3.2 Higher Order Interpolation Spaces.- 3.3 Estimates for KA.- 3.4 Maximal Regularity.- 4 Maximal Sobolev Regularity.- 4.1 Temperate Distributions.- 4.2 Fourier Transforms and Convolutions.- 4.3 The Hilbert Transform.- 4.4 UMD Spaces and Fourier Multipliers.- 4.5 Properties of UMD Spaces.- 4.6 Fractional Powers.- 4.7 Bounded Imaginary Powers.- 4.8 Perturbation Theorems.- 4.9 Sums of Closed Operators.- 4.10 Maximal Regularity.- IV Variable Domains.- 1 Higher Regularity.- 1.1 Properties of Differentiable Functions.- 1.2 General Solvability Results for Cauchy Problems.- 1.3 Estimates for Evolution Operators.- 1.4 Evolution Operators on Interpolation Spaces.- 1.5 The Cauchy Problem.- 2 Constant Interpolation Spaces.- 2.1 Semigroup and Convergence Estimates.- 2.2 Assumptions and Consequences.- 2.3 Construction of Evolution Operators.- 2.4 Estimates for Evolution Operators.- 2.5 The Cauchy Problem.- 2.6 Abstract Boundary Value Problems.- 3 Maximal Regularity.- 3.1 Abstract Initial Boundary Value Problems.- 3.2 Isomorphism Theorems.- V Scales of Banach Spaces.- 1 Banach Scales.- 1.1 General Concepts.- 1.2 Power Scales.- 1.3 Extrapolation Spaces.- 1.4 Dual Scales.- 1.5 Interpolation-Extrapolation Scales.- 2 Evolution Equations in Banach Scales.- 2.1 Semigroups in Interpolation-Extrapolation Scales.- 2.2 Parabolic Evolution Equations in Banach Scales.- 2.3 Duality.- 2.4 Approximation Theorems.- 2.5 Final Value Problems.- 2.6 Weak Solutions and Duality.- 2.7 Positivity.- 2.8 General Evolution Equations.- List of Symbols.