Well-posedness of the Muskat problem with initial data

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• 作者：C.H. Arthur Cheng^a ; ^{cchsiao@math.ncu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Rafael Granero-Belinchó ; n^b ; ^{rgranero@math.ucdavis.edu" class="auth_mail" title="E-mail the corresponding author} ; Steve Shkoller^b ; ^{shkoller@math.ucdavis.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35R35 ; 35Q35 ; 35S10 ; 76B03
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：2 January 2016
• 年：2016
• 卷：286
• 期：Complete
• 页码：32-104
• 全文大小：800 K

文摘

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium   for small class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003357&_mathId=si1.gif&_user=111111111&_pii=S0001870815003357&_rdoc=1&_issn=00018708&md5=13a26b9324c3c3a3c21efd627fef9c59" title="Click to view the MathML source">H²class="mathContainer hidden">class="mathCode">$H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815003357&_mathId=si1.gif&_user=111111111&_pii=S0001870815003357&_rdoc=1&_issn=00018708&md5=13a26b9324c3c3a3c21efd627fef9c59" title="Click to view the MathML source">H²class="mathContainer hidden">class="mathCode">$H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.