# Preprint 2006-019

*L*^{2} semigroup and linear
stability for Riemann solutions of conservation laws

## Xiao-Biao Lin

**Abstract:**
Riemann solutions for the systems of conservation laws
*u*_{τ}+*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
*u _{t}*+(D

*f*(

*u*)−

*x*I)

*u*=0. For the linear variational system around the Riemann solution with

_{x}*n*-Lax shocks, we introduce a semigroup in the Hilbert space with weighted

*L*

^{2}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*(e

^{γt}) 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

*u*+(D

_{t}*f*(

*u*)−

*x*I)

*u*=ε

_{x}*u*.

_{xx}