Gearhart–Prüss Theorem and linear stability for Riemann solutions of conservation laws
Abstract: We study the spectral and linear stability of the Riemann solutions with multiple Lax shocks for systems of conservation laws uτ+f(u)ξ=0. Using the self-similar change of variables x=ξ/τ, t=ln(τ), Riemann solutions become stationary to the system ut+(Df(u)−xI)ux=0. In the space of O((1+|x|)−η) functions, we show that if Re λ>−η, then λ is either an eigenvalue or resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, there are resonance values where the determinant can be arbitrarily small but nonzero. A C0 semigroup is constructed. Using the Gearhart–Prüss Theorem, we show that the solutions are of O(eγt) if γ is greater than the largest real parts of the eigenvalues and the resonance values. We study examples where Riemann solutions have two or three Lax-shocks.