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# L2 semigroup and linear stability for Riemann solutions of conservation laws

Abstract: Riemann solutions for the systems of conservation laws uτ+f(u)ξ=0 are self-similar solutions of the form u=u(ξ/τ). Using the change of variables x=ξ/τ, t=ln(τ), Riemann solutions become stationary to the system ut+(Df(u)−xI)ux=0. For the linear variational system around the Riemann solution with n-Lax shocks, we introduce a semigroup in the Hilbert space with weighted L2 norm. We show that (A) the region Re λ>−η consists of normal points only. (B) Eigenvalues of the linear system correspond to zeros of the determinant of a transcendental matrix. They lie on vertical lines in the complex plane. There can be resonance values where the response of the system to forcing terms can be arbitrarily large, see Definition \ref{resonance}. Resonance values also lie on vertical lines in the complex plane. (C) Solutions of the linear system are O(eγt) for any constant γ that is greater than the largest real parts of the eigenvalues and the coordinates of resonance lines. This work can be applied to the linear and nonlinear stability of Riemann solutions of conservation laws and the stability of nearby solutions of the Dafermos regularizations ut+(Df(u)−xI)uxuxx.

Paper:
Available from the journal web page (see below).
Author(s):
Xiao-Biao Lin,
Publishing information:
Dynamics of PDE 2, No.4, 301–333 (2005)