The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effects * *Research 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.

Abstract

The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension N ⩾ 2. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the optimal time decay rates for the constructed global solutions.