Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

Haibo Cui; Qunyi Bie; Zheng-An Yao

doi:10.3934/dcdsb.2018156

Article Contents

2018, Volume 23, Issue 4: 1395-1410. Doi: 10.3934/dcdsb.2018156

This issue Previous Article On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems Next Article Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain

Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

1.
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2.
College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
3.
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author
* Corresponding author

Received: July 31, 2016

Revised: January 31, 2018

Early access: April 2018

Published: April 2018

Research Supported by the NNSF of China (Grant Nos. 11601164, 11271381 and 11701325), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002), the Natural Science Foundation of Fujian Province of China (Grant Nos. 2016J05010 and 2017J05007) and the Scientific Research Funds of Huaqiao University (Grant No.15BS201).

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Abstract

This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase flow model in $\mathbb{R}^N\,(N≥2)$ . We concentrate on the critical Besov spaces based on the $L^p$ setting. We improve the range of Lebesgue exponent $p$ , for which the system is locally well-posed, compared to [22]. Applying Lagrangian coordinates is the key to our statements, as it enables us to obtain the result by means of Banach fixed point theorem.

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Mathematics Subject Classification: Primary: 35Q35, 76N10.

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