Strong solutions for the compressible barotropic fluid model of Korteweg type in the bounded domain

This paper is concerned with a barotropic model of capillary compressible fluids derived by Dunn and Serrin (1985). We consider the initial boundary value problem in bounded domains of Rd(d=2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d (d=2,3)$$\end{document} with more general pressure including Van der Waals equation of state. First, the local in time existence and uniqueness of strong solutions is proved for initial data belonging to the Sobolev space W22×W21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2}_{2}\times W^{1}_{2}$$\end{document}, relying on a new linearization technique and the contraction mapping principle. Next, we show that there exists a global unique strong solution by the continuation argument of local solution under small initial perturbation. The proof is based on the elementary energy method, but with a new development, where some techniques are introduced to establish the uniform energy estimates and to treat the pressure function with non-increasing property.