{"title":"On $ L^1 $ estimates of solutions of compressible viscoelastic system","authors":"Y. Ishigaki","doi":"10.3934/dcds.2021174","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to <inline-formula><tex-math id=\"M2\">\\begin{document}$ W^{2,1} $\\end{document}</tex-math></inline-formula>, and is sufficiently small in <inline-formula><tex-math id=\"M3\">\\begin{document}$ H^4\\cap L^1 $\\end{document}</tex-math></inline-formula>, the solutions grow in time at the same rate as <inline-formula><tex-math id=\"M4\">\\begin{document}$ t^{\\frac{1}{2}} $\\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id=\"M5\">\\begin{document}$ L^1 $\\end{document}</tex-math></inline-formula> due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"46 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Continuous Dynamical Systems - S","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/dcds.2021174","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to \begin{document}$ W^{2,1} $\end{document}, and is sufficiently small in \begin{document}$ H^4\cap L^1 $\end{document}, the solutions grow in time at the same rate as \begin{document}$ t^{\frac{1}{2}} $\end{document} in \begin{document}$ L^1 $\end{document} due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.
We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to \begin{document}$ W^{2,1} $\end{document}, and is sufficiently small in \begin{document}$ H^4\cap L^1 $\end{document}, the solutions grow in time at the same rate as \begin{document}$ t^{\frac{1}{2}} $\end{document} in \begin{document}$ L^1 $\end{document} due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.
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