Stability of undercompressive viscous shock profiles of hyperbolic–parabolic systems

Abstract

Extending to systems of hyperbolic-parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in L p , 1 p ∞.

Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves

“…The recent paper [LZ2], on the other hand, features a uniform basin of attraction as the shock amplitude goes to zero, something that is not attained from the basic stability argument given in Section 3; we discuss this issue further in Appendix B. Note that the results of [SX,LZ2] apply only to smallamplitude "Lax type" waves and artificial (Laplacian) viscosity, whereas our results in [Z1,MaZ4,Z4,HZ,RZ] apply in principle to waves of arbitrary amplitude and type and physical (partially parabolic) viscosity; see [HLZ, BHZ] for proofs of stability in the large-amplitude limit for isentropic gas dynamics and MHD.…”

“…In the semilinear case considered here, Corollary 3.7 could be proved in a more straightforward fashion by a contraction-mapping argument applied directly to the system (3.20)-(3.21), bypassing the continuous induction argument above. However, in more delicate situations such as the quasilinear parabolic or hyperbolic-parabolic cases, it is advantageous for reasons of regularity to separate the issues of short-time existence/well-posedness and long-time bounds, as we have done here; see [MaZ2,MaZ4,Z4,RZ] for further discussion.…”

Conditional stability of unstable viscous shocks

“…Another interesting problem would be to extend our conditional stability result to the case of nonclassical under-or overcompressive shocks using pointwise estimates as in [10,18]; see Remark 4.7.…”

“…We point out that the finite-dimensional part z of v is in fact controlled pointwise by its L 2 norm and thus by |w| H 2 , satisfying |z(x, t)| Ce −θ|x| |z(·, t)| L 2 (x) Ce −θ|x| |w(·, t)| 2 H 2 (x) , making possible a pointwise version of the argument above. This is a key point in treating the nonclassical over-or undercompressive cases, which appear to require pointwise bounds [10,18]. …”

Existence and Stability of Viscoelastic Shock Profiles

“…In this paper, generalizing work of Antman and Malek-Madani [AM] in the incompressible shear flow case, we carry out the numerical and analytical study of the existence and stability of planar viscoelastic traveling waves in a 3d solid, for a simple prototypical elastic energy density, both for the general compressible and the incompressible shear flow case. We establish that the resulting equations fall into the class of symmetrizable hyperbolicparabolic systems studied in [MaZ2,MaZ3,MaZ4,RZ,Z4], hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. This important point was previously left undecided, due to a lack of the necessary abstract stability framework.…”