Abstract by Bryn Balls-Barker
Spectral Stability of Ideal-Gas Shock Layers in the Strong Shock Limit
An open question in gas dynamics is the stability of viscous shock layers, or traveling-wave solutions of the compressible Navier-Stokes equations. In general, the Evans function, which is typically computed numerically, plays a key role in determining the stability of these traveling wave solutions.
The goal of this research is to analytically describe the spectral stability of ideal-gas shock layers in the strong shock limit using the Evans function. The numerical stability of this system has been previously demonstrated (Humpherys et al. 2009) and we seek to make this stability more rigorous with an analytic proof. We do this by analytically solving for a basis of the unstable and stable manifolds and then by using these solutions to create the Evans function. Due to numerical instability in the Evans system associated with the compressible Navier-Stokes equations, we use a change of variables to find the bases and then use energy estimates to bound the potentially unstable eigenvalues. With the resulting analytic approximation to the Evans function, we are able to study meaningful bounds on the stability of the shock layers.