Large-time behavior of smooth solutions to the isothermal compressible fluid models of Korteweg type with large initial data

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• 作者：Zhengzheng Chen ^{chenzzandu@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q35 ; 35L65 ; 35B40
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：144
• 期：Complete
• 页码：139-156
• 全文大小：740 K

文摘

This paper is concerned with the large-time behavior of smooth non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with the viscosity coefficient pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si1.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=51ee9e237ccdc299796e98c9a721b7ad" title="Click to view the MathML source">μ(ρ)=ρ^αpan>pan class="mathContainer hidden">pan class="mathCode">$\mu \left(\rho \right)={\rho }^{\alpha }$pan>pan>pan> and the capillarity coefficient pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si2.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=6a1a9f4c334a58903e51cecde6f19875" title="Click to view the MathML source">κ(ρ)=ρ^βpan>pan class="mathContainer hidden">pan class="mathCode">$\kappa \left(\rho \right)={\rho }^{\beta }$pan>pan>pan>. Here pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si3.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=ca96c4e738d4c5f48f5f156fa65bb785" title="Click to view the MathML source">α∈Rpan>pan class="mathContainer hidden">pan class="mathCode">$\alpha \in \mathbb{R}$pan>pan>pan> and pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si4.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=aac0fbc8fd3ddec84100e545549a4ce2" title="Click to view the MathML source">β∈Rpan>pan class="mathContainer hidden">pan class="mathCode">$\beta \in \mathbb{R}$pan>pan>pan> are some parameters. Depending on whether the far-fields of the initial data are the same or not, we prove that the corresponding Cauchy problem admits a unique global smooth solution which tends to constant states or rarefaction waves respectively, as time goes to infinity, provided that pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si5.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=69bac81643560e5f0d0dfd345218c946" title="Click to view the MathML source">αpan>pan class="mathContainer hidden">pan class="mathCode">$\alpha$pan>pan>pan> and pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=eff981fb34f13b9fd7d3c321d6d84027" title="Click to view the MathML source">βpan>pan class="mathContainer hidden">pan class="mathCode">$\beta$pan>pan>pan> satisfy some conditions. Note that the initial perturbation can be arbitrarily large. The proofs are given by the elementary energy method and Kanel’s technique (Kanel, 1968). Compared with former results in this direction obtained by Germain and LeFloch (2016), and Chen et al. (2015), the main novelties of this paper lie in the following: First, we obtain the global existence of smooth solutions with large data for some new varieties of parameters pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si5.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=69bac81643560e5f0d0dfd345218c946" title="Click to view the MathML source">αpan>pan class="mathContainer hidden">pan class="mathCode">$\alpha$pan>pan>pan> and pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301432&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301432&_rdoc=1&_issn=0362546X&md5=eff981fb34f13b9fd7d3c321d6d84027" title="Click to view the MathML source">βpan>pan class="mathContainer hidden">pan class="mathCode">$\beta$pan>pan>pan>. Second, the large-time behavior of smooth large solutions around constant states is established.