Paradifferential Operators and Conormal Distributions

by Robin Spratte
Doctoral thesis
Date of Examination:2019-10-28
Date of issue:2020-02-24
Advisor:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Dorothea Bahns
Persistent Address: http://dx.doi.org/10.53846/goediss-7879

 

 

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Abstract

English

In this thesis we develop a generalization of Hörmander’s symbol calculus of conor- mal distributions [Hö07, Chapter 18.2] and provide techniques for applications to nonlinear hyperbolic Partial Differential Equations. In particular we will provide explicit expansion formulas for symbols of conormal distributions under multiplica- tion (Theorem 2.16 and Corollary 2.25) and nonlinear superposition with Hölder- Zygmund continuous functions (Theorem 2.39). We also define the class of diffeomorphisms of conormal type and establish their structure as a group (Theorems 2.28 and 2.41), again giving explicit expansion formulas for their symbols. This enables us to define conormal distributions with respect to non smooth hypersurfaces endowed with the established symbol calculus. The definitions we give and the methods we develop are applicable to nonlinear Partial Differential Equations. In Chapter 3 we explicitly construct approximate symbolic solutions to a Cauchy problem with coefficients and datum given as conormal distributions. We obtain solvability of the reduced problem within a sufficiently smooth remainder space. In Chapter 4 we provide propagation of conormality for the developed symbolic calculus under hyperbolic quasilinear equations of first order.
Keywords: PDE; Nonlinear PDE; Microlocal Analysis; Symbol Calculus; Conormal Distributions; Paradifferential Operators; Pseudodifferential Operators
 
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