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.

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.

Time Decay Rate of Solutions Toward the Viscous Shock Waves under Periodic Perturbations for the Scalar Conservation Law with Nonlinear Viscosity

In this paper,it is proved that the weak solution to the Cauchy problem for the scalar viscous conservation law,with nonlinear viscosity,different far field states and periodic perturbations,not only exists globally in time,but also converges towards the viscous shock wave of the corresponding Riemann problem as time goes to infinity.Furthermore,the decay rate is shown.The proof is given by a technical energy method.

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 axe 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 pro- files 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.