Large-time behavior of smooth solutions to the isothermal compressible fluid models of Korteweg type with large initial data

This paper is concerned with the large-time behavior of smooth non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with the viscosity coefficient μ(ρ)=ραμ(ρ)=ρα and the capillarity coefficient κ(ρ)=ρβκ(ρ)=ρβ. Here α∈Rα∈R and β∈Rβ∈R are some parameters. Depending on whether the far-fields of the initial data are the same or not, we prove that the corresponding Cauchy problem admits a unique global smooth solution which tends to constant states or rarefaction waves respectively, as time goes to infinity, provided that αα and ββ satisfy some conditions. Note that the initial perturbation can be arbitrarily large. The proofs are given by the elementary energy method and Kanel’s technique (Kanel, 1968). Compared with former results in this direction obtained by Germain and LeFloch (2016), and Chen et al. (2015), the main novelties of this paper lie in the following: First, we obtain the global existence of smooth solutions with large data for some new varieties of parameters αα and ββ. Second, the large-time behavior of smooth large solutions around constant states is established.