Global stability of combination of a viscous contact wave with rarefaction waves for the compressible fluid models of Korteweg type

Abstract

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density-and temperaturedependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of the viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the heat conductivity coefficient satisfy some smallness conditions. The proof is based on some refined L 2 -energy estimates to control the possible growth of the solutions caused by the highly nonlinearity of the system, the interactions of waves from different families and large data, and the key ingredient is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.

Asymptotic stability of the superposition of viscous contact wave with rarefaction waves for the compressible Navier-Stokes-Maxwell equations

“…This new technique was needed by Zeng in [41] to obtain the stability of the superposition of shock waves with contact discontinuities for systems of viscous conservation laws, and was also used by Huang-Wang in [11] for the global stability of the same wave patterns and system as in [7]. In addition, some results about the asymptotic stability of the composite wave as in [7] were also shown for some more complex models, and we refer interested readers to [36], [5], [24], [1], [28], and references therein. Due to the frameworks of viscous shock wave and rarefaction wave are not compatible with each other, the timeasymptotic stability of the combination of viscous shock wave and rarefaction wave is still an interesting and challenging question for the Navier-Stokes equations!…”

Stability of viscous contact wave for the full compressible Navier–Stokes–Korteweg equations with large perturbation

“…In [9], two types of global solvability results were obtained for systems where the viscosity, capillarity and heat conductivity coefficients satisfy certain conditions. Moreover, in a recent study, Chen and Sheng [10] showed that when µ = µ(v, θ), κ = κ(v, θ), α = α(v, θ), κ θθ (v, θ) < 0, and |κ θ (v, θ)| ≪ 1, the combination of a viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if γ − 1 is sufficiently small.…”

“…The purpose of this work is to study the mathematical behaviour of such compressible viscous capillary fluids in R. The Korteweg model [10] we consider here in Lagrangian coordinates reads as…”

Low Mach number limit of the full compressible Navier-Stokes-Korteweg equations with general initial data

“…Kotschote [24,25,26] established the local existence, global existence and time-asymptotics of strong solution for the non-isentropic compressible NSK equations in a bounded domain with C 3 -boundary. About the stability of basic nonlinear wave patterns such as the discontinuity wave, viscous contact wave and the rarefaction wave of one dimensional compressible NSK equations, we can refer to [6,7,8,31,37] and the references therein.…”