$L^p$ asymptotic behavior of perturbed viscous shock profiles

Details $L^p$ asymptotic behavior of perturbed viscous shock profiles $L^p$ asymptotic behavior of perturbed viscous shock profiles

2004-08-17

arxiv.org/abs/math/0408227v1

Mathematics

Analysis of PDEs

59 pp

Scientific paper

We investigate the $L^p $ asymptotic behavior $(1\le p \le \infty)$ of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier--Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e., the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in $L^p$.

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