Stability of Contact Discontinuity with General Perturbation for the Compressible Navier-Stokes Equations with Reaction Diffusion

In this article,for the compressible Navier-Stokes equations which have reaction diffusion, the stability of contact discontinuities is considered. The new characteristic for the flow is appearance of the divergence between energy gained and lost because of the reaction . In the energy equations,the term related to the mass fraction of the reactant leads to new technical problem. To solve this problem, in terms of the solutions,a new system should be set up. Using the anti-derivative method and the elaborated energy method, we obtain that as long as the general perturbation of the initial datum plane and the strength of the contact wave are properly small, the contact wave is nonlinear and stable. As a byproduct, we can establish the convergence velocity of contact wave.

Stability of the composite wave for compressible Navier–Stokes/Allen–Cahn system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier–Stokes/Allen–Cahn system which is a combination of the classical Navier–Stokes system with an Allen–Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of [Formula: see text] for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier–Stokes/Allen–Cahn system. The proof is mainly based on a basic energy method.

Asymptotic stability of viscous contact wave and rarefaction waves for the system of heat-conductive ideal gas without viscosity

This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system has the solution consisting of a contact discontinuity and rarefaction waves, we show that if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends to the corresponding composition of a viscous contact wave with rarefaction waves, which extended the results by Huang–Li–Matsumura [Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system, Arch. Ration. Mech. Anal. 197 (2010) 89–116.], where they treated the viscous and heat-conductive ideal gas.