Local existence for the non-resistive MHD equations in Besov spaces

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• 作者：Jean-Yves Chemin^a ; ^{chemin@ann.jussieu.fr" class="auth_mail" title="E-mail the corresponding author} ; David S. McCormick^b ; ¹ ; ² ; ^{mccormick.d.s@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; James C. Robinson^b ; ³ ; ^{j.c.robinson@warwick.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Jose L. Rodrigo^b ; ⁴ ; ^{j.l.rodrigo@warwick.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Besov spaces ; Magnetohydrodynamics ; MHD
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：2 January 2016
• 年：2016
• 卷：286
• 期：Complete
• 页码：1-31
• 全文大小：478 K

文摘

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si1.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=75678dc3dbbcb368deded48231c293cc" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si12.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=fbeb5e9572a6ebe3d62d3f7cfef40b41" title="Click to view the MathML source">n=2,3class="mathContainer hidden">class="mathCode">$n=2,3$, for divergence-free initial data in certain Besov spaces, namely class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si3.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=a6807bb410f8ca5a418bd805f95dac7f">class="imgLazyJSB inlineImage" height="27" width="88" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815003424-si3.gif">class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si4.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=419eb9c8ea0c56f13d6181b2cb570cb8">class="imgLazyJSB inlineImage" height="27" width="75" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815003424-si4.gif">class="mathContainer hidden">class="mathCode">. The a priori estimates include the term class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si5.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=cee6de82f0b530b591eafccf2b7e6c74">class="imgLazyJSB inlineImage" height="25" width="115" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815003424-si5.gif">class="mathContainer hidden">class="mathCode"> on the right-hand side, which thus requires an auxiliary bound in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si6.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=2d497e2057b9acf17882cc6a942b0e7b" title="Click to view the MathML source">H^n/2−1class="mathContainer hidden">class="mathCode">$H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003424&_mathId=si7.gif&_user=111111111&_pii=S0001870815003424&_rdoc=1&_issn=00018708&md5=0209e37e31b0ba7cb39b78a73e458f9f" title="Click to view the MathML source">H^1/2class="mathContainer hidden">class="mathCode">$H^{1/2}$ is required, which we prove using the splitting method of Calderón (1990) [2]. By contrast, our proof that such solutions are unique only applies to the 3D case.