Global strong solution of 2D Navier–Stokes–Korteweg system

Abstract

In this paper, we consider the Cauchy problem to the two‐dimensional Navier–Stokes–Korteweg system with nonvacuum and general pressure laws and establish the global well‐posedness of strong solution for general initial data in the framework of Sobolev spaces.

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

“…Subsequently, Haspot 18 continued to investigate the Cauchy problem of the system ()–() with and established the global solutions under the setting of slightly subcritical L p type initial data, where the specific choice of the pressure is crucial since it provides a gain of integrability on the effective velocity. Following the assumptions on the viscosity coefficients of Haspot, 18 Yu and Wu 19 established the global well‐posedness of strong solution to 2D NSK system with nonvacuum and general pressure laws in the framework of Sobolev spaces. Chikami and Kobayashi 20 established the global solutions to the compressible NSK system under linear stability conditions in critical Besov spaces and obtained the optimal decay rates of the global solutions in the ‐framework.…”