Linearized stability of extreme shock profiles in systems of conservation laws with viscosity

Pego, Robert L.

doi:10.1090/S0002-9947-1983-0716831-9

Linearized stability of extreme shock profiles in systems of conservation laws with viscosity
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by Robert L. Pego PDF

Trans. Amer. Math. Soc. 280 (1983), 431-461 Request permission

Abstract:

For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, ${u_t} + f{(u)_x} = \varepsilon {u_{xx}}$, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight ${e^{cx}}$ is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.

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