Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

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• 作者：Peter Constantin^a ; ^{const@math.princeton.edu" class="auth_mail" title="E-mail the corresponding author} ; Igor Kukavica^b ; ^{kukavica@usc.edu" class="auth_mail" title="E-mail the corresponding author} ; Vlad Vicol^a ; ^{vvicol@math.princeton.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q35 ; 35Q30 ; 76D09
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：November-December 2016
• 年：2016
• 卷：33
• 期：6
• 页码：1569-1588
• 全文大小：376 K

文摘

We consider the incompressible Euler equations on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000669&_mathId=si1.gif&_user=111111111&_pii=S0294144915000669&_rdoc=1&_issn=02941449&md5=726a2f7dbafbec3703a484c8652e27ce" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000669&_mathId=si2.gif&_user=111111111&_pii=S0294144915000669&_rdoc=1&_issn=02941449&md5=492be71637de168cb8088aac4c0ed05d" title="Click to view the MathML source">T^dclass="mathContainer hidden">class="mathCode">${\mathbb{T}}^d$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000669&_mathId=si12.gif&_user=111111111&_pii=S0294144915000669&_rdoc=1&_issn=02941449&md5=a84fd0d53055809c867a2910d7ba330e" title="Click to view the MathML source">d∈{2,3}class="mathContainer hidden">class="mathCode">$d\in \{2,3\}$. We prove that:

(a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius).

(b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces  , e.g. Gevrey-class regularity in the label class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000669&_mathId=si4.gif&_user=111111111&_pii=S0294144915000669&_rdoc=1&_issn=02941449&md5=cc3b75844c59f3c500946e105b02a79f" title="Click to view the MathML source">a₁class="mathContainer hidden">class="mathCode">$a_1$ and Sobolev regularity in the labels class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000669&_mathId=si5.gif&_user=111111111&_pii=S0294144915000669&_rdoc=1&_issn=02941449&md5=393ac1da023f05c84ad1512f0d60d231" title="Click to view the MathML source">a₂,…,a_dclass="mathContainer hidden">class="mathCode">$a_2,\dots ,a_d$.

(c) In Eulerian coordinates both results (a) and (b) above are false.