Stability Of Viscous Shock Waves Research Articles

Nonlinear Stability of Large Amplitude Viscous Shock Waves to the One-dimensional System of Viscoelasticity

In this paper, we introduce a new effective velocity to study the nonlinear stability of viscous shock waves to the one-dimensional system of viscoelasticity. Any viscous shock wave of arbitrary am...

Pointwise stability of reaction diffusion fronts

AbstractUsing pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential convergence in weightedLpand Sobolev norms, while capturing the new feature of spatial diffusion at Gaussian rate. Novel features of the argument are a pointwise Green function decomposition reconciling spectral decomposition and short-time Nash-Aronson estimates and an instantaneous tracking scheme similar to that used in the study of stability of viscous shock waves.

Nonlinear stability of viscous shock waves for one-dimensional nonisentropic compressible Navier–Stokes equations with a class of large initial perturbation

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small. The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a $2 × 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21, 23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction wave, are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of rarefaction wave fan to compressible Euler equations is proved to 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in [18] sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves

In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun [Instantaneous shock location ...] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give by a rescaling argument a simple treatment of nonlinear stability in the small-amplitude case.

Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier–Stokes equations with density dependent viscous coefficient

Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier–Stokes equations with density dependent viscous coefficient

Stability of viscous shock waves for the one-dimensional compressible navier-stokes equations with density-dependent viscosity

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel' and the continuation argument.

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

We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instantaneous location of a viscous shock. We show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Guès, Métivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection.

Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

An efficient shooting algorithm for Evans function calculations in large systems

In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products [J.C. Alexander, R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World 2 (4) (1995) 471–507; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Ph.D. Thesis, Indiana University, Bloomington, 1998; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (235) (2001) 1071–1088; L.Q. Brin, K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, in: Seventh Workshop on Partial Differential Equations, Part I, 2001, Rio de Janeiro, Mat. Contemp. 22 (2002) 19–32; T.J. Bridges, G. Derks, G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Physica D 172 (1–4) (2002) 190–216]. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as ( n k ) , where n is the dimension of the system written as a first-order ODE and k (typically ∼ n / 2 ) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing “pure” (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury–Davey [A. Davey, An automatic orthonormalization method for solving stiff boundary value problems, J. Comput. Phys. 51 (2) (1983) 343–356; L.O. Drury, Numerical solution of Orr-Sommerfeld-type equations, J. Comput. Phys. 37 (1) (1980) 133–139] and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.

Stability of undercompressive shock profiles

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under–overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations | u 0 ( x ) | ⩽ E 0 ( 1 + | x | ) − 3 / 2 in C 0 + α , E 0 sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in [K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741–871], extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in [A. Szepessy, Z. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal. 122 (1993) 53–103; T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (11) (1997) 1113–1182; K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741–871; K. Zumbrun, Refined wave-tracking and nonlinear stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747–768; M.-R. Raoofi, L 1 -asymptotic behavior of perturbed viscous shock profiles, thesis, Indiana Univ., 2004; C. Mascia, K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (4) (2002) 773–904; C. Mascia, K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic–parabolic systems, Comm. Pure Appl. Math. 57 (7) (2004) 841–876; C. Mascia, K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal. 169 (3) (2003) 177–263; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of hyperbolic–parabolic systems, Arch. Ration. Mech. Anal. 172 (1) (2004) 93–131; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., in press], and for special “weakly coupled” (respectively scalar diffusive–dispersive) undercompressive profiles in [T.P. Liu, K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys. 168 (1) (1995) 163–186; T.P. Liu, K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. 174 (2) (1995) 319–345] (respectively [P. Howard, K. Zumbrun, Pointwise estimates for dispersive–diffusive shock waves, Arch. Ration. Mech. Anal. 155 (2000) 85–169]). In particular, together with spectral results of [K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity–capillarity, SIAM J. Appl. Math. 60 (2000) 1913–1924], our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model [M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 81 (4) (1983) 301–315] for van der Waal gas dynamics or elasticity with viscosity–capillarity.

Errata to: 'Pointwise semigroup methods and stability of viscous shock waves'

Errata to: 'Pointwise semigroup methods and stability of viscous shock waves'

Analytically varying eigenvectors and the stability of viscous shock waves

Analytically varying eigenvectors and the stability of viscous shock waves

Stability of Viscous Shock Waves for Problems with Nonsymmetric Viscosity Matrices

Sufficient conditions for nonlinear stability of viscous shock wave solutions of systems of conservation laws are given. The analysis applies to strong shocks of Lax type but is restricted to perturbations with zero mass. We use the Laplace transform and reduce the question of stability to a spectral condition for the resolvent equation of the linearized problem.

Numerical testing of the stability of viscous shock waves

A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability. Accuracy of the method is demonstrated through comparison against exact solutions, a convergence study, and evaluation of approximate error equations. Robustness is demonstrated by applying the method to waves for which no current analytic results apply (highly nonlinear waves from the cubic model and strong shocks from gas dynamics). An interesting aspect of the analysis is the need to incorporate features from the analytic Evans function theory for purposes of numerical stability. For example, we find it necessary, for numerical accuracy, to solve ODEs on the space of wedge products.

Stability of Viscous Shock Waves¶Associated with Non-Convex Modes

In this paper we consider small-amplitude viscous shock waves associated with a possibly non-convex mode (i.e., a not necessarily “genuinely nonlinear field”) and prove their time-asymptotic stability towards sufficiently small perturbations of not necessarily vanishing mass. We thus extend the result of [SX] to the case of non-convex modes and the result of [F1] to the case of non-zero mass perturbations.

Pointwise semigroup methods and stability of viscous shock waves

Considered as rest points of ODE on L p , stationary vis- cous shock waves present a critical case for which standard semigroup methods do not suce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact that precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ((Sat), (K.2), resp.), each of which can be reduced to the standard semi- group setting by Sattinger's method of weighted norms. We overcome this diculty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard (H.1), Kapit- ula (K.1-2), and Zeng (Ze,LZe). These techniques allow us to do \hard analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, sucient for the analysis of linear and nonlinear stability. The method is general, and should nd applications also in other situations of sensitive stability. Central to our analysis is a notion of \eective point spectrum that can be extended to regions of essential spectrum. This turns out to be intimately related to the Evans function, a well-known tool for the spectral analysis of traveling waves. Indeed, crucial to our whole analy- sis is the \Gap Lemma of (GZ,KS), a technical result developed origi- nally in the context of Evans function theory. Using these new tools, we can treat general over- and undercompressive, and even strong shock waves for systems within the same framework used for standard weak (i.e. slowly varying) Lax waves. In all cases, we show that stability is determined by the simple and numerically computable condition that the number of zeroes of the Evans function in the right complex half- plane be equal to the dimension of the stationary manifold of nearby traveling wave solutions. Interpreting this criterion in the conserva-

A Remark on the Stability of Viscous Shock Waves

Recently, Szepessy and Xin gave a new proof of stability of viscous shock waves. A curious aspect of their argument is a possible disturbance of zero mass, but $\log (t)t^{{{ - 1} /2}} $ amplitude in the vicinity of the shock wave. This would represent a previously unobserved phenomenon. However, only an upper bound is established in their proof. Here, we present an example of a system for which this phenomenon can be verified by explicit calculation. The disturbance near the shock is shown to be precisely of order $t^{{{ - 1} /2}} $ in amplitude.

Nonlinear stability of viscous shock waves

Nonlinear stability of viscous shock waves