Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

Sheng

2019

Nonlinearity

Self Cite

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.

Section: Introductionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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

Sheng

2019

Nonlinearity

Self Cite

“…They shown the global existence of weak solutions which may contain vacuum in the whole space R. Moreover, the global existence of non-vacuum weak solutions was also obtained on both the torus and the whole space in [17]. Finally, for the existence and nonlinear stability of some elementary waves (such as rarefaction waves, viscous shock profiles and contact discontinuity wave) to the isothermal or non-isothermal compressible fluid models of Korteweg type with small initial data, we refer to [22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 94%

Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

Journal of Differential Equations

Chai

Dong

et al. 2015

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This paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with density-dependent viscosity coefficient and capillarity coefficient. The case when the viscosity coefficient μ(ρ) = ρ α and the capillarity coefficient κ(ρ) = ρ β for some parameters α, β ∈ R is considered. Under some conditions on α, β, we first show the global existence of large solutions around constant states if the far-fields of the initial data are the same, while if the far-fields of the initial data are different, we prove the global stability of rarefaction waves with large strength. Here global stability means the initial perturbation can be arbitrarily large. Our analysis is based on the elementary energy method and the technique developed by Y. Kanel' [29].

“…The global existence and large-time behavior of solutions to the compressible fluid models of Korteweg type have been studied by many authors. For small initial data, we refer to [6,7] for the global existence of smooth solutions around constant states in Sobolev space, [10,11,12,13,17,18,20,19] for the large-time behavior of smooth solutions in Sobolev space, and [5,9] for the global existence and uniqueness of strong solutions in Besov space.…”

Section: Introductionmentioning

confidence: 99%

Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

Zhao

2017

Z. Angew. Math. Phys.

Self Cite

The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type have been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient µ(ρ) = ρ α , the capillarity coefficient κ(ρ) = ρ β , and the heat-conductivity coefficientα(θ) = θ λ for some parameters α, β, λ ∈ R is considered. Under some assumptions on α, β and λ, we prove the global existence and time-asymptotic behavior of large solutions around constant states. The proofs are given by the elementary energy method combined with the technique developed by Y. Kanel' [23] and the maximum principle.