Global Well-Posedness and Decay Rates of Strong Solutions to a Non-Conservative Compressible Two-Fluid Model

被引:0
作者
Steinar Evje
Wenjun Wang
Huanyao Wen
机构
[1] University of Stavanger,Department of Petroleum Engineering
[2] University of Shanghai for Science and Technology,College of Science
[3] South China University of Technology,Department of Mathematics
来源
Archive for Rational Mechanics and Analysis | 2016年 / 221卷
关键词
Decay Rate; Capillary Pressure; Global Existence; Strong Solution; Global Weak Solution;
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学科分类号
摘要
In this paper, we consider a compressible two-fluid model with constant viscosity coefficients and unequal pressure functions P+≠P-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P^+\neq P^-}$$\end{document}. As mentioned in the seminal work by Bresch, Desjardins, et al. (Arch Rational Mech Anal 196:599–629, 2010) for the compressible two-fluid model, where P+=P-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P^+=P^-}$$\end{document} (common pressure) is used and capillarity effects are accounted for in terms of a third-order derivative of density, the case of constant viscosity coefficients cannot be handled in their settings. Besides, their analysis relies on a special choice for the density-dependent viscosity [refer also to another reference (Commun Math Phys 309:737–755, 2012) by Bresch, Huang and Li for a study of the same model in one dimension but without capillarity effects]. In this work, we obtain the global solution and its optimal decay rate (in time) with constant viscosity coefficients and some smallness assumptions. In particular, capillary pressure is taken into account in the sense that ΔP=P+-P-=f≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta P=P^+ - P^-=f\neq 0}$$\end{document} where the difference function f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}$$\end{document} is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to P-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P^-}$$\end{document}. This assumption plays an key role in the analysis and appears to have an essential stabilization effect on the model in question.
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页码:1285 / 1316
页数:31
相关论文
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