Evolution Equations in Scales of Banach Spaces

1. Front Matter

   Pages 1-12

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2. Introduction

   • Oliver Caps

   Pages 13-25

3. Tools from functional analysis

   • Oliver Caps

   Pages 27-77

4. Well-posedness of the time-dependent linear Cauchy problem

   • Oliver Caps

   Pages 78-129

5. Quasilinear Evolution Equations

   • Oliver Caps

   Pages 130-165

6. Applications to linear, time-dependent evolution equations

   • Oliver Caps

   Pages 166-244

7. Applications to quasilinear evolution equations

   • Oliver Caps

   Pages 245-294

8. Back Matter

   Pages 295-309

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The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. The usual functional analytic methods for studying evolution equations are formu­ lated within the setting of unbounded, closed operators in one Banach space. This setting is not adapted very well to the study of many pseudo differential and differential equations because these operators are naturally not given as closed, unbounded operators in one Banach space but as continuous opera­ tors in a scale of function spaces. Thus, applications within the setting of unbounded, closed operators require a considerable amount of additional work because one has to construct suitable closed realizations of these operators. This choice of closed realizations is technically complicated even for simple applications. The main feature of the new functional analytic approach of the book is to study the operators in scales of Banach spaces that are constructed by simple reference operators. This is a natural setting for many operators acting in scales of function spaces. The operators are only expected to respect the scale and to satisfy certain inequalities but we can avoid completely the choice of any closed realization of these operators which is of great importance in applications. We use the mapping properties of the reference operators to prove sufficient conditions for well-posedness of linear and quasilinear Cauchy problems. In the linear, time-dependent case these conditions are shown to characterize well-posedness. A similar result in the standard setting (i. e.

• Applications to quasilinear evolution equations
• Quasilinear evolution equations
• Tools from functional analysis
• Well-posedness
• functional analysis
• linear Cauchy problem

• Mainz, Deutschland

  Oliver Caps

Dr. Oliver Caps, Universität Mainz

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