Stabilität elastodynamischer Schockwellen
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In this dissertation we consider planar shock-waves both in isentropic gas dynamics and in elastodynamics in two space dimensions. In the theory of hyperbolic systems, shock-waves are represented by weak solutions to nonlinear systems of conservation laws, satisfying jump conditions on a smooth hypersurface as well as entropy conditions motivated by physics. Given a reference shock-solution of the conservation laws and a multidimensional perturbation of the initial data or a small wave impinging on the shock-front, one may ask for the stability of the shock, i.e. if the shock-structure persists under perturbation. Due to the fundamental work of A. Majda and A. M. Blokhin, the question regarding nonlinear stability can be answered by analyzing a stability function known as the Lopatinski-determinant. Considering the equations of isentropic elastodynamics in eulerian coordinates, we aim to give explicit expressions of stability-separatrices using the Lopatinski-determinant. The latter equations are supplemented by additional divergence constraints allowing to write them as a symmetric hyperbolic system at the cost of losing the conservative form. By considering the so-called beta-model which was firstly proposed by Barker, Monteiro & Zumbrun in 2021 for MHD-LAX-shocks, we are able to present for the first time an explicit expression of the Lopatinski-determinant for compressive LAX-shocks in isentropic inviscid elastodynamics for general deformations. And by monotonicity arguments on specific intervals a way to calculate the boundaries of the stability domains analytically.
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SCHLIPF, Markus, 2023. Stabilität elastodynamischer Schockwellen [Dissertation]. Konstanz: University of KonstanzBibTex
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