L^pASYMPTOTIC BEHAVIOR OF PERTURBED VISCOUS SHOCK PROFILES

Abstract

We investigate the L p asymptotic behavior (1 ≤ p ≤ ∞) 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 .

Galloping instability of viscous shock waves

“…Thus, x-and y-derivatives do not always bring down additional additional factors t −1/2 of temporal decay as for constant-coefficient Gaussians, but may instead, through derivatives falling on spatially-dependent parameters, bring down factors e −η|x| or e −η|y| of spatial decay; see [ZH,Z4] for further discussion. The neglected term (3.38), which is harmless for the stability analyses of [R,MaZ2,MaZ4,Z1], may be recognized as a term of this form.…”

Stability of undercompressive shock profiles

“…It may be readily checked numerically, as described, e.g., in [5][6][7][8][9]. It was shown by various techniques in [38][39][40][41][42]45,54,55] that the linearized stability condition (D) is also sufficient for nonlinear orbital stability of Lax or overcompressive profiles of arbitrary amplitude. However, up to now, this result had not been verified in the undercompressive case.…”

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

“…The same Evans assumption has already been shown to imply long time stability of viscous profiles in the 1D case in [KK] for zero mass perturbations and [Z2,MaZ1,MaZ2,MaZ3,MaZ4,MaZ5,Z3,HZ,Ra] for general perturbations, and in the multidimensional case in [Z1, Z3, Z4] (for general perturbations); see also the important groundwork of [GZ, ZH, ZS] and [K1,K2,KS,LZe,H1,H2]. A treatment of the scalar multidimensional case (for which the Evans assumption always holds, by the maximum principle) may be found in [HoZ2,HoZ3].…”

“…The endgame of [Z1] could be described rather as "parabolic": integration on the parabolic contour Γ(ξ ) reveals an additional temporal decay due to diffusion that is essential to the proof of nonlinear stability in dimensions less than or equal to two. (Note: In dimension one, somewhat further care is needed; specifically, translation of the shock must be projected out [Z2,MaZ1,MaZ2,MaZ3,MaZ4,MaZ5,Z3,HZ,Ra]. )…”