Abstract. In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier-Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which smallamplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength.By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we

We present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the KdV-KS equation modeling weakly unstable flow of a thin fluid film on an incline in the small-amplitude KdV limit. The argument proceeds by verification of a stability condition derived by Bar-Nepomnyashchy and Johnson-Noble-Rodrigues-Zumbrun involving inner products of various elliptic functions arising through the KdV equation. One key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time. Another is the way of handling analytic interpolation in several variables by a two-step process carving up the parameter space into manageable pieces for rigorous evaluation. These and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability.

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and MalekMadani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic-parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.

gas dynamics and Freistühler and Trakhinin for compressible magnetohydrodynamics, we study by a combination of asymptotic ODE estimates and numerical Evans function computations the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a γ -law gas with γ ∈ [1, 3]. Other γ -values may be treated similarly, but were not checked numerically. Depending on magnetic field strength, these shocks may be of fast Lax, intermediate (overcompressive), or slow Lax type; however, the shock layer is independent of magnetic field, consisting of a purely gas-dynamical profile. In each case, our results indicate stability. Interesting features of the analysis are the need to renormalize the Evans function in order to pass continuously across parameter values where the shock changes type or toward the large-amplitude limit at frequency λ = 0 and the systematic use of winding number computations on Riemann surfaces.