On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems
On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems
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DOI：
10.1081/pde-120020499
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发表时间：
2003-01
影响因子：
1.9
通讯作者：
D. Bresch;B. Desjardins;Chi-Kun Lin
中科院分区：
文献类型：
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作者：
D. Bresch;B. Desjardins;Chi-Kun Lin
本文给出了Dunn和Serrin在[1]Dunn Je，Serrin J.关于间隙工作热力学的毛细可压缩流体的等温模型的一些数学结果。拱形理性机甲肛门。1985；88(2)：95-133)，可作为相变模型。我们考虑周期区域Ω=Td(d=2ou 3)或条形区域Ω=(0，1)×Td−1。我们考察了粘度μ和毛细系数κ对密度ρ的依赖关系。根据我们考虑的情况，会得到不同的结果。例如，对于一个粘性μ(ρ)=νρ和一个表面张力，我们证明了在没有数据小假设的情况下，Korteweg系统弱解的整体存在性。该模型包括浅水模型和润滑模型。我们讨论了浅水方程的结果的有效性，因为密度比Korteweg情形的密度不那么规则。
Abstract In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in [1]Dunn JE, Serrin J. On the thermodynamics of interstitial working. Arch Rational Mech Anal. 1985; 88(2):95–133), which can be used as a phase transition model. We consider a periodic domain Ω = T d (d = 2 ou 3) or a strip domain Ω = (0,1) × T d −1. We look at the dependence of the viscosity μ and the capillarity coefficient κwith respect to the density ρ. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity μ(ρ) = νρ and a surface tension the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.