# Preprint 2007-006

# Gearhart–Prüss Theorem and linear stability for Riemann solutions of conservation laws

## Xiao-Biao Lin

**Abstract:**
We study the spectral and linear stability
of the Riemann solutions with multiple Lax shocks
for systems of conservation laws
*u*_{τ}+*f*(*u*)_{ξ}=0.
Using the self-similar change of variables
*x*=ξ/τ, *t*=ln(τ),
Riemann solutions become stationary to the system
*u*_{t}+(*Df*(*u*)−*xI*)*u*_{x}=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 *C*^{0}
semigroup is constructed.
Using the Gearhart–Prüss Theorem,
we show that the solutions are of
*O*(e^{γt}) 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.