{"title":"Blowup for $ {{\\rm{C}}}^{1} $ solutions of Euler equations in $ {{\\rm{R}}}^{N} $ with the second inertia functional of reference","authors":"Manwai Yuen","doi":"10.3934/math.2023412","DOIUrl":null,"url":null,"abstract":"<abstract><p>The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference:</p> <p><disp-formula> <label/> <tex-math id=\"FE1\"> \\begin{document}$ { H}_{ref}{ (t) = }\\frac{1}{2}\\int_{\\Omega(t)}\\left( { \\rho-\\bar{\\rho}}\\right) \\left\\vert { \\vec{x} }\\right\\vert ^{2}dV{{ , }} $\\end{document} </tex-math></disp-formula></p> <p>for the blowup phenomena of $ C^{1} $ solutions $ (\\rho, \\vec{u}) $ with the support of $ \\left({ \\rho-\\bar{\\rho}}, \\vec{u}\\right) $, and with a positive constant $ { \\bar{\\rho}} $ for the adiabatic index $ \\gamma > 1 $. We find that if the total reference mass</p> <p><disp-formula> <label/> <tex-math id=\"FE2\"> \\begin{document}$ M_{ref}(0) = { \\int_{{\\bf R}^{N}}} (\\rho_{0}({ \\vec{x}})-\\bar{\\rho})dV\\geq0, $\\end{document} </tex-math></disp-formula></p> <p>and the total reference energy</p> <p><disp-formula> <label/> <tex-math id=\"FE3\"> \\begin{document}$ E_{ref}(0) = \\int_{{\\bf R}^{N}}\\left( \\frac{1}{2}\\rho_{0}({ \\vec {x}})\\left\\vert \\vec{u}_{0}({ \\vec{x}})\\right\\vert ^{2}+\\frac {K}{\\gamma-1}\\left( \\rho_{0}^{\\gamma}({ \\vec{x}})-\\bar{\\rho }^{\\gamma}\\right) \\right) dV, $\\end{document} </tex-math></disp-formula></p> <p>with a positive constant $ K $ is sufficiently large, then the corresponding solution blows up on or before any finite time $ T > 0 $.</p></abstract>","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"AIMS Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/math.2023412","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference:
for the blowup phenomena of $ C^{1} $ solutions $ (\rho, \vec{u}) $ with the support of $ \left({ \rho-\bar{\rho}}, \vec{u}\right) $, and with a positive constant $ { \bar{\rho}} $ for the adiabatic index $ \gamma > 1 $. We find that if the total reference mass
The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference: \begin{document}$ { H}_{ref}{ (t) = }\frac{1}{2}\int_{\Omega(t)}\left( { \rho-\bar{\rho}}\right) \left\vert { \vec{x} }\right\vert ^{2}dV{{ , }} $\end{document} for the blowup phenomena of $ C^{1} $ solutions $ (\rho, \vec{u}) $ with the support of $ \left({ \rho-\bar{\rho}}, \vec{u}\right) $, and with a positive constant $ { \bar{\rho}} $ for the adiabatic index $ \gamma > 1 $. We find that if the total reference mass \begin{document}$ M_{ref}(0) = { \int_{{\bf R}^{N}}} (\rho_{0}({ \vec{x}})-\bar{\rho})dV\geq0, $\end{document} and the total reference energy \begin{document}$ E_{ref}(0) = \int_{{\bf R}^{N}}\left( \frac{1}{2}\rho_{0}({ \vec {x}})\left\vert \vec{u}_{0}({ \vec{x}})\right\vert ^{2}+\frac {K}{\gamma-1}\left( \rho_{0}^{\gamma}({ \vec{x}})-\bar{\rho }^{\gamma}\right) \right) dV, $\end{document} with a positive constant $ K $ is sufficiently large, then the corresponding solution blows up on or before any finite time $ T > 0 $.
期刊介绍:
AIMS Mathematics is an international Open Access journal devoted to publishing peer-reviewed, high quality, original papers in all fields of mathematics. We publish the following article types: original research articles, reviews, editorials, letters, and conference reports.