Viscous Conservation Laws Research Articles

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t +f(u) x =μu xx with the initial datau 0 which tend to the constant statesu ± asx→±∞. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f′(u ±). Moreover, the rate of asymptotics in time is investigated. For the casef′(u+)<s<f′(u−), if the integral of the initial disturbance over (−∞,x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate ast→∞. This rate seems to be almost optimal compared with the rate in the casef=u 2/2 for which an explicit form of the solution exists. The rate is also obtained in the casef′(u ± =s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

Hyperbolic conservation laws with stiff relaxation terms and entropy

AbstractWe study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley &amp; Sons, Inc.

Stability of travelling waves for non‐convex scalar viscous conservation laws

AbstractTravelling waves of a viscous conservation law (so‐called viscous profiles) are shown to be stable in polynomialy weighted L∞ spaces. There is no assumption of convexity on the nonlinear term and thus earlier results of Il'in and Oleǐnik, Sattinger, and Kawashima and Matsumura are generalized. The method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula. Estimates are derived for the semigroup through a new technique for estimating the resolvent. © 1993 John Wiley &amp; Sons, Inc.

Nonlinear stability of overcompresive shock waves in a rotationally invariant system of viscous conservation laws

This paper proves that certain non-classical shock waves in a rotationally invariant system of viscous conservation laws posses nonlinear large-time stability against sufficiently small perturbations. The result applies to small intermediate magnetohydrodynamic shocks in the presence of dissipation.

A subcell resolution method for viscous systems of conservation laws

We consider the generalization of scalar subcell resolution schemes to systems of viscous conservation laws. For this purpose we use a weakly nonlinear geometrical optics approximation for parabolic perturbations of hyperbolic conservation laws and the Roe-type field by field decomposition. Computations of the reactive Navier-Stokes equations are presented as an application.

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.

Computation of viscous or nonviscous conservation law by domain decomposition based on asymptotic analysis

AbstractAn accurate and efficient numerical method has been developed for a nonlinear diffusion convection‐dominated problem. The scheme combines asymptotic methods with usual solution techniques for hyperbolic problems. After having localized shock or corner layers and rescaling, first terms of the inner expansion are computed. Using the same concepts gives a method to compute a very accurate solution of the nonlinear conservation law. Because our numerical scheme is based on a uniform approximation throughout the domain, the shock is localized very accurately and there is practically no smearing out. Numerical computations are presented. Another novel feature is the ability to break down the problem according to subdomains of different local behavior, based on asymptotic analysis, which may make it feasible to do computations with different processors.

A Numerical Study of Riemann Problem Solutions and Stability for a System of Viscous Conservation Laws of Mixed Type

A numerical study of the isothermal fluid equations with a nonmonotone equation of state (like that of van der Waals) and with viscosity and capillarity terms is presented. This system is ill-posed (i.e., elliptic in x vs. t) in some regions of state space and well-posed (i.e., hyperbolic) in other regions. Thus, it may serve as a model for describing dynamic phase transitions. Numerical computations of phase jumps, shock waves, and rarefaction waves for this system are presented. Although the solution of the Riemann problem is not unique, all of these waves are found to be stable to infinitesimal initial perturbations. Criteria are found for instability after $O( 1 )$ initial perturbations. An analytic argument is made for stability of phase transitions.

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the self-similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves isO(t −3/4+1/2p+σ) inL p , 1≦p≦∞, with σ>0 being arbitrarily small.

Asymptotic Stability of Planar Rarefaction Waves for Viscous Conservation Laws in Several Dimensions

Asymptotic Stability of Planar Rarefaction Waves for Viscous Conservation Laws in Several Dimensions

Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions

This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary L 2 {L^2} -energy method.

Convergence of diffusion waves of solutions for viscous conservation laws

Convergence of diffusion waves of solutions for viscous conservation laws

Asymptotic stability of rarefaction waves for 2 × 2 viscous hyperbolic conservation laws—The two-modes case

In this paper, we continue our study on the asymptotic behavior toward rare-faction waves of a general 2 × 2 system of hyperbolic conservation laws with positive viscosity matrix. It is shown that when the initial data is a small perturbation of a weak rarefaction wave (a linear superposition of a 1-rarefaction wave and a 2-rarefaction wave) for the corresponding inviscid hyperbolic conservation laws, then the solution of the Cauchy problem for the viscous system globally exists and tends to the rarefaction wave. The result is proved by using an energy method, combining the technique in [Z. P. Xin, J. Differential Equations 73 (1988), 45–77], and using the characteristic-energy method of T. P. Liu [ Mem. Amer. Math. Soc. 328 (1975), 1–108].

Stability of Viscous Scalar Shock Fronts in Several Dimensions

We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation.

Stability of viscous scalar shock fronts in several dimensions

We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation.

Decay to Rarefactions for Scalar Multidimensional Viscous Conservation Laws

We show that solutions to the viscous conservation laws in several space dimensions converge to rarefactions as time tends to infinity, when the initial data belongs to some large global class of functions. The convergence takes place in L ∞ (R n ), and in L P (R×K), where K is compact in R n−1 , and we determine the asymptotic rate

Asymptotic stability of rarefaction waves for 2 ∗ 2 viscous hyperbolic conservation laws

This paper concerns the asymptotic behavior toward rarefaction waves of the solution of a general 2 × 2 hyperbolic conservation laws with positive viscosity. We prove that if the initial data is close to a constant state and its values at ±∞ lie on the kth rarefaction curve for the corresponding hyperbolic conservation laws, then the solution tends as t → ∞ to the rarefaction wave determined by these states.

Stability of shock waves for the Broadwell equations

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].

Convergence to diffusion waves of solutions for viscous conservation laws

We study the large-time behavior of solutions of viscous conservation laws. It is shown that solutions tend to diffusion waves, which are constructed based on the heat equation and Burgers equation. The convergence is in theLp, 1≦p≦∞ sense and is obtained as a consequence of theL2 decay of the difference between the solution and its asymptotic state of diffusion waves.

Shock waves for compressible navier‐stokes equations are stable

AbstractIt is shown that shock waves for the compressible Navier‐Stokes equations are nonlinearly stable. A perturbation of a shock wave tends to the shock wave, properly translated in phase, as time tends to infinity. Through the consideration of conservation of mass, momentum and energy we obtain an a priori estimate of the amount of translation of the shock wave and the strength of the linear and nonlinear diffusion waves that arise due to the perturbation. Our techniques include the energy method for parabolic‐hyperbolic systems, the decomposition of waves, and the energy‐characteristic method for viscous conservation laws introduced earlier by the author.