Title: Bifurcation and stability of viscous shock waves in viscous conservation laws
Language: English
Authors: Achleitner, Franz Georg 
Qualification level: Doctoral
Keywords: Evans Funktion; viskose Erhaltungsgleichungen; viskose Schockwellen; spektrale Stabilität; Melnikov Funktion
Evans function; viscous conservation laws; viscous shock waves; spectral stability; Melnikov function
Advisor: Szmolyan, Peter
Assisting Advisor: Freistühler, Heinrich 
Issue Date: 2009
Number of Pages: 157
Qualification level: Doctoral
It is a natural idea to study the stability of shock waves by analyzing the spectrum of the linearized evolution operator. The Evans function approach to such problems provides a general geometric framework to study and exploit spectral properties of the linearized problem. Briefly speaking, the Evans function is an analytic function of the spectral parameter whose zeros to the right of the essential spectrum correspond to eigenvalues. A shock wave is spectrally stable, if the spectrum of the related linear operator consists of eigenvalues with negative real part and the eigenvalue zero. Zumbrun and collaborators have shown that spectral stability of viscous shock wave implies its nonlinear stability.
We study the generic case of a saddle-node bifurcation of viscous shock waves, where the family of viscous shock waves can be described via the Melnikov function. By relating the derivatives of the Melnikov function with derivatives of the Evans function, we prove a change of stability within the family of viscous shock waves. We apply our results to an example in magnetohydrodynamics.
URI: https://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-26281
Library ID: AC05040093
Organisation: E101 - Institut für Analysis und Scientific Computing 
Publication Type: Thesis
Appears in Collections:Thesis

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