L² semigroup and linear stability for Riemann solutions of conservation laws

Riemann solutions for the systems of conservation laws $u_\tau + f(u)_\xi = 0$are self-similar solutions of the form $u=u(\xi/\tau)$. Using the change of variables$x = \xi/\tau, t= ln(\tau)$, Riemann solutions become stationary to the system$u_t + (Df(u) - x I) u_x = 0$. For the linear variational system around the Riemannsolution with $n$-Lax shocks, we introduce a semigroup in the Hilbert space withweighted $L^2$ norm. We show that (A) the region $\Re \lambda> -\eta$ consistsof normal points only. (B) Eigenvalues of the linear system correspond to zeros ofthe determinant of a transcendental matrix. They lie on vertical lines in the complexplane. There can be {\em resonance values} where the response of the system toforcing terms can be arbitrarily large, see Definition \ref{resonance}. Resonancevalues also lie on vertical lines in the complex plane. (C) Solutions of the linear systemare $O(e^{\gamma t})$ for any constant $\gamma$ that is greater than the largest real partsof the eigenvalues and the coordinates of {\em resonance lines}. This work can beapplied to the linear and nonlinear stability of Riemann solutions of conservation lawsand the stability of nearby solutions of the Dafermos regularizations $u_t + (Df(u) - x I) u_x =\epsilon u_{xx}$.

Keywords

L² semigroup, conservation laws, multiple shocks, linear stability, eigenvalue and resonance values

2010 Mathematics Subject Classification

Primary 46-xx. Secondary 35-xx.

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Published 1 January 2005