On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

Fuyi Xu; Meiling Chi; Lishan Liu; Yonghong Wu

doi:10.3934/dcds.2020140

Article Contents

2020, Volume 40, Issue 5: 2515-2559. Doi: 10.3934/dcds.2020140

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On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

1.
School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, Shandong, China
2.
School of Mathematical Science, Qufu Normal University, Qufu, 263516, Shandong, China
3.
Department of Mathematics and Statistics, Curtin University, Perth, 6845, WA, Australia

^* Corresponding author: Fuyi Xu
^* Corresponding author: Fuyi Xu

Received: October 2018

Revised: July 2019

Published: March 2020

The first author is supported by the National Natural Science Foundation of China (11501332, 11771043, 11871302, 51976112), the Natural Science Foundation of Shandong Province (ZR2015AL007), and Young Scholars Research Fund of Shandong University of Technology

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Abstract

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove 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 time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

Keywords:

Well-posedness,
decay rates,
non-conservative viscous compressible two-fluid system,
Besov spaces.

Mathematics Subject Classification: Primary: 76T10, 76N10.

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References

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