Global existence and time decay estimate of solutions to the compressible Navier–Stokes–Korteweg system under critical condition

Article type: Research Article

Authors: Kobayashi, Takayuki | Tsuda, Kazuyuki^{; *}

Affiliations: Osaka University, 1-3, Machikaneyamacho, Toyonakashi, 560-8531, Japan. E-mail: jtsuda@sigmath.es.osaka-u.ac.jp

Correspondence: [*] Corresponding author. E-mail: jtsuda@sigmath.es.osaka-u.ac.jp.

Abstract: In this research, we study the global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global L2 solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part.

Keywords: Partial differential equations, compressible Navier–Stokes–Korteweg system, global solution, time decay rate

DOI: 10.3233/ASY-201600

Journal: Asymptotic Analysis, vol. 121, no. 2, pp. 195-217, 2021