Stability Of Viscous Shock Profiles Research Articles

Asymptotic stability of viscous shock profiles for the 1D compressible Navier–Stokes–Korteweg system with boundary effect

This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. The analysis is based on the elementary $L^2$-energy method and the key point is to deal with the boundary estimates.

Nonlinear stability of viscous shock profiles for compressible Navier–Stokes equations with temperature-dependent transport coefficients and large initial perturbation

We study the nonlinear stability of viscous shock profiles of the Cauchy problem to the one-dimensional compressible Navier–Stokes equations for a viscous and heat-conducting ideal polytropic gas whose transport coefficients depend on both the density and temperature. The viscous shock profiles are shown to be asymptotically stable under large initial perturbation when the adiabatic exponent $$\gamma $$ is suitably close to 1, and the strengths of the viscous shock profiles are sufficiently small which allow initial density to have large oscillation. The crucial step in the energy estimates is to obtain uniformly positive lower and upper bounds of density.

Multi-dimensional Wave Propagation Over a Burgers Shock Profile

We study the stability of viscous shock profiles under multi-dimensional perturbation in order to understand the propagation ofmulti-dimensional dispersionwaves over the compressive shock profile. Our analysis is based on the new algorithm of explicitly constructing the Green’s function for the system linearized around the shock profile. We first construct Green’s function for the system linearized around the inviscid profile by matching the two half space problems. The Green’s function around the shock profile is then constructed by iterations. Our approach is of a general nature. We carry out the approach for a simple model possessing the Burgers shocks and wave structure similar to acoustic cones in multi-dimensional gas flows. The approach exhibits a rich phenomenon of wave propagation along and dispersing away from the shock profile.

Numerical proof of stability of viscous shock profiles

We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number [Formula: see text], below which profiles are known by energy estimates to be stable, to the upper stability boundary of [Formula: see text], above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.

One-dimensional compressible Navier-Stokes equations with large density oscillation

This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [6, 7] is excluded, such a problem has been studied in [5, 11] and local stability of weak viscous shock profiles is well-established, but for the corresponding result with large initial perturbation, fewer results have been obtained. Our main purpose is to deduce the corresponding nonlinear stability result with large initial perturbation by exploiting the elementary energy method. As a first step toward this goal, we show in this paper that for certain class of ``large initial perturbation which can allow the initial density to have large oscillation, similar stability result still holds. Our analysis is based on the continuation argument and the technique developed by Kanel' in [4].

Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity

For the two-dimensional Navier-Stokes equations of isentropic magnetohydrodynamics (MHD) with γ-law gas equation of state, γ≤ 1, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes. In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. Considered as three-dimensional solutions, undercompressive shocks are Lax-type (Alfven) waves. For the monatomic and diatomic cases γ = 5/3 and γ = 7/5, with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well, both with respect to two-dimensional and (in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.

Spectral Stability of Small-Amplitude Viscous Shock Waves in Several Space Dimensions

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function \({\mathcal{E}(\rho\lambda,\rho\omega)}\) which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of \({\mathcal{E}(\rho\lambda,\rho\omega)}\) and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength \({\epsilon}\) of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality.

Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation

The Evans function theory, which has recently been applied to the study of linear stability of viscous shock profiles, is developed below for semi-linear relaxation. We study the linear stability of shock profiles in the Lax, undercompressive and overcompressive cases. The results we obtain are similar to those found for viscous approximations by Gardner and Zumbrun [Commun. Pure Appl. Math. 51 (7) (1998) 797].

Stability of Steady Solutions of Burgers's Equation

We develop tools to discuss the stability of viscous shock profiles for conservation laws. The technique is based on replacing the far field regions by boundary conditions that describe the effect of the outer solution on the solution in the shock layer region. We apply the technique to Burgers's equation and prove stability for perturbations that are not localized in space.

Stability of systems of viscous conservation laws

We consider systems of conservation laws and give conditions for nonlinear stability of viscous shock profiles. The analysis applies to classical shocks of arbitrary strength. © 1998 John Wiley & Sons, Inc.

On the linearized stability of viscous shock profiles for systems of conservation laws

This paper is concerned with the linearized stability of traveling wave solutions for systems of viscous hyperbolic conservation laws. The main purpose is to show that for a given traveling wave with shock profile from any characteristic family, there exists an appropriate weighted norm space such that the traveling wave is exponentially stable in this space. As a consequence, if the initial disturbance has average zero and decays exponentially fast as ¦x¦ → ∞, then the corresponding solution of the linearized equation decays to zero exponentially fast in t on any compact interval in x. The proof is given by applying an elementary weighted characteristic energy method to the integrated linearized system, based on the underlying wave structure.

Nonlinear asymptotic stability of viscous shock profiles for conservation laws

Nonlinear asymptotic stability of viscous shock profiles for conservation laws