On the Cauchy problem for a class of degenerate hyperbolic equations

by Matthias Krüger
Doctoral thesis
Date of Examination:2018-05-18
Date of issue:2018-08-31
Advisor:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Dorothea Bahns
Persistent Address: http://dx.doi.org/10.53846/goediss-7038

 

 

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Abstract

English

In this thesis, a pseudodifferential calculus for a degenerate hyperbolic Cauchy problem is developed. The model for this problem originates from a certain observation in fluid mechanics, and is then extended to a more general class of hyperbolic Cauchy problems where the coefficients degenerate like a power of $t + |x|^2$ as $(t,x) \to (0,0)$. Symbol classes and pseudodifferential operators are introduced. In this process, it becomes apparent that exactly in the origin, these operators are of type (1,1). Although these operators are not $L^2$-continuous in general, a proof of continuity in $\mathscr C([0,T],L^2(\mathbb R^d))$ is given for a suitable subclass. An adapted scale of function spaces is defined, where at $t = 0$ these spaces coincide with 2-microlocal Sobolev spaces with respect to the Lagrangian $\dot T^*_0\mathbb R^d$. In these spaces, energy estimates are derived, so that a symbolic approach can be applied to prove wellposedness of the Cauchy problem.
Keywords: degenerate hyperbolic Cauchy problem; partial differential equations; pseudodifferential operators
 
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