Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

The unique global solvability of multi-dimensional compressible Navier–Stokes–Poisson–Korteweg model

Pages: 2169 – 2190



Fuyi Xu (School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, China)

Yeping Li (Department of Mathematics, East China University of Science and Technology, Shanghai, China)


The present paper is dedicated to the study of the Cauchy problem for compressible Navier–Stokes–Poisson–Korteweg model in any dimension $d \geq 2$, which simultaneously involves the lower order potential term and the higher order capillarity term. The unique global solvability of the system is obtained when the initial data are close to a stable equilibrium state in a functional setting invariant by the scaling of the associated equations. In particular, one may construct the unique global solution for a class of large highly oscillating initial velocities in physical dimensions $d=2,3$.


unique global solvability, highly oscillating velocity, compressible Navier–Stokes–Poisson–Korteweg model, critical Besov spaces

2010 Mathematics Subject Classification

35Mxx, 35Q35

Fuyi Xu is partially supported by the National Natural Science Foundation of China (11501332,11771043,51976112), the Natural Science Foundation of Shandong Province (ZR2015AL007), and Young Scholars Research Fund of Shandong University of Technology. Yeping Li is partially supported by the National Natural Science Foundation of China (11671134).

Received 7 October 2019

Accepted 7 June 2020

Published 22 December 2020