Asymptotic stability of viscous contact wave for the inflow problem of the heat-conductive ideal gas without viscosity

This paper is devoted to studying the inflow problem governed by the non-viscous and heat-conductive gas dynamic system in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact wave. The contact discontinuity in the linearly degenerate field is less stable, and the dissipative mechanism for non-viscous systems is also weaker than that of viscous systems, these all make the problem more challenging. We used the weighted energy estimates to overcome those difficulties. Some technical discussions were created carefully by taking good advantage of properties of the supersonic region and the viscous contact wave.     

Asymptotic stability of viscous contact wave to a radiation hydrodynamic limit model

This paper is concerned with the large time behavior of the solutions for 1D radiation hydrodynamic limit model without viscosity and its asymptotic stability of the viscous contact discontinuity wave under the smallness assumption of the strength of the contact wave and initial perturbations. The present pressure includes a fourth-order term about the absolute temperature from radiation effect which brings the main difficulty. Furthermore, the dissipative of the system is weaker for the lack of viscosity. All these make the problem more challenging. The prove is mainly based on the energy method, including normal and radial directions energy estimates.     

Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws

In this paper, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves with viscous contact waves for systems of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation.     

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non‐isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial‐boundary value problem of the non‐isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero‐order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability of viscous contact wave for compressible navier-stokes system of general gas with free boundary

In this paper, we study the large time behavior of solutions to the nonisentropic Navier-Stokes equations of general gas, where polytropic gas is included as a special case, with a free boundary. First we construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation, in finite time as the heat conductivity tends to zero. Then we prove the viscous contact wave is asymptotic stable if the initial perturbations and the strength of the contact wave are small. This generalizes our previous result [6] which is only for polytropic gas.     

Asymptotic stability of viscous contact wave for the one-dimensional compressible viscous gas with radiation

In this paper, we study the large-time behavior of solutions of the Cauchy problem to a one-dimensional Navier-Stokes-Poisson coupled system, modeling the dynamics of a viscous gas in the presence of radiation. When the far field states are suitably given, and the corresponding Riemann problem for the Euler system admits only a contact discontinuity wave solution with the far field states as Riemann initial data. Then, we can define a “viscous contact wave” for such a Navier-Stokes-Poisson coupled system. Based on elementary energy methods and ellipticity of the equation of the radiation flux, we can prove the “viscous contact wave” is stable provided the strength of the contact discontinuity wave and the perturbation of the initial data are suitably small.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts

For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.     

The gap lemma and geometric criteria for instability of viscous shock profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3-19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.     

Pointwise Green function bounds and stability of combustion waves

Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier-Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman-Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu-Ying and Tesei-Tan for Majda's model and the reactive Navier-Stokes equations, respectively.     

Stability of a transonic profile arising from divergent detonations

We establish the existence of viscous profile of an undriven divergent detonation wave. The structure of the wave is a viscous shock followed by a reaction zone containing a sonic point. So the divergent detonation wave profile is a transonic profile.

We further establish the existence of classical global solutions of the Cauchy problem. The proof consists of establishing a priori estimates for the solution by a maximum principle and using Hölder estimates for solutions of parabolic equations. Finally, the nonlinear stability of the viscous transonic profile is established. The main reason for the stability is that there is a damping term due to the divergent nature of the problem.     

Asymptotic behavior toward the combination of contact discontinuity with rarefaction waves for 1-D compressible viscous gas with radiation

This paper is concerned with the large-time behavior toward the combination of two rarefaction waves and viscous contact wave for the Cauchy problem to a one-dimensional Navier–Stokes–Poisson coupled system, modeling the dynamics of a viscous gas in the presence of radiation. We show that the composite wave with small strength is asymptotically stable under partially large initial perturbations. The proofs are based on the more refined energy estimates to control the possible growth of the perturbations induced by two different waves and large data.     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength 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. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

Global stability of viscous contact wave for 1-D compressible Navier–Stokes equations

The viscous contact waves for one-dimensional compressible Navier–Stokes equations has recently been shown to be asymptotically stable. The stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not. Up to now, local stability results toward viscous contact waves of compressible Navier–Stokes equations have been well established (see Huang et al., 2006, 2008, 2009 [9], [10], [7]), but there are few results for the global stability in the case of Cauchy problem which is the purpose of this paper. The proof is based on an elementary energy method using an inequality concerning the heat kernel (see Lemma 1 of Huang et al., 2010 [7]).     

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L²‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of viscous shock wave under periodic perturbation for compressible Navier–Stokes–Korteweg system

In this paper, we study the stability of a viscous shock wave for the isentropic Navier–Stokes–Korteweg (N-S-K) equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.     

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.     

Asymptotic stability of the viscous shock wave for the one-dimensional compressible Navier–Stokes equations with density dependent viscous coefficient on the half space

Archiv der Mathematik - In this paper, taking the boundary effect into consideration, we investigate the asymptotic stability of a viscous shock wave for one-dimensional isentropic compressible...     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.