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

Abstract

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].

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

“…Therefore, in this paper, we study the existence and uniqueness of smooth solutions in Sobolev spaces for the compressible NSK system without increasing pressure laws. Finally, we point out there are many recent studies for the asymptotic behaviors of solutions to the Cauchy problem of one dimensional compressible NSK system with different states at infinity x =± ∞ , see Tan and Guo, Tan and Zhang, Tang and Zhang, Tang and Zhang, Thanh et al, and Tsuda and the reference therein.…”

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

“…While if g(α, β) = 0, the estimate of I 2 is divided into three cases. Since the estimates are the same as those in [21], we cite the results directly. According to the estimates (2.22)-(2.24) in [21], we have…”

“…• Compared to the isothermal case [21], a new case (i): α = 0, β = −2 is obtained in Theorem 1.1 for the parameters α and β by some more delicate analysis (see the proof of Lemmas 2.3, 2.6 and 2.11 for details). We believe that some similar results can also be established for the isothermal fluid by employing the method in this paper, which will be reported in a forthcoming paper by the authors.…”

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

“…Recently, there are lots of literatures on the well‐posedness and large‐time behavior of global solutions for the compressible Navier‐Stokes‐Poisson equations and the compressible Navier‐Stokes‐Korteweg equations; the interested reader can refer to previous studies . As we known, the quasi‐neutral limit λ→0 is the important problem for the hydrodynamic model from plasma and semiconductors.…”

“…Recently, there are lots of literatures on the well-posedness and large-time behavior of global solutions for the compressible Navier-Stokes-Poisson equations and the compressible Navier-Stokes-Korteweg equations; the interested reader can refer to previous studies. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] As we known, the quasi-neutral limit λ→0 is the important problem for the hydrodynamic model from plasma and semiconductors. For the quasi-neutral limit in the Euler-Poisson equation and the Naviver-Stokes-Poisson equation, we can refer to previous studies [22][23][24][25][26][27][28][29][30][31] and the therein theorem.…”