Note on the weak-strong uniqueness criterion for the β-QG in Morrey-Campanato space

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• 作者：Saddek Gala ; ^a ; ^{sadek.gala@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{sgala@ksu.edu.sa" class="auth_mail" title="E-mail the corresponding author} ; ^{gala.sadek@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Maria Alessandra Ragusa^b ; ^{maragusa@dmi.unict.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quasi-geostrophic equation ; Weak&ndash ; strong uniqueness criterion ; Morrey&ndash ; Campanato space
• 刊名：Applied Mathematics and Computation
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：293
• 期：Complete
• 页码：65-71
• 全文大小：326 K

文摘

Consider the β  -quasi-geostrophic equations with the initial data rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si1.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=574fcf83bdfe335998475550c9ca9da8" title="Click to view the MathML source">θ₀∈L²(R²)r hidden">roll">row>θ0∈L2row>(riant="double-struck">R2)row>row>. Let θ   and rc">rce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si2.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=8138d08e3fa64d0d1cfbd2b662ffd1c0">rce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si2.gif">ript>rder="0" style="vertical-align:bottom" width="13" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0096300316305069-si2.gif">ript>r hidden">roll">r accent="true">θ˜r> be two weak solutions with the same initial value θ₀. If θ satisfies the energy inequality and ifwhere rc">rce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si4.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=8fccb2dfb34549ba34236fa1d434a2e1">rce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si4.gif">ript>rder="0" style="vertical-align:bottom" width="79" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0096300316305069-si4.gif">ript>r hidden">roll">row>r>riant="script">M.r>row>p,qrow>row>(riant="double-struck">R2)row>row> denotes the homogeneous Morrey–Campanato space, then we have rc">rce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si5.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=80ff903a36e47a957c4560d3dafeca7d">rce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si5.gif">ript>rder="0" style="vertical-align:bottom" width="49" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0096300316305069-si5.gif">ript>r hidden">roll">row>θ=r accent="true">θ˜r>row>. Due to the embedding relation rc">rce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si6.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=483c425b37dcbad7162f0e43f9851a2b">rce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si6.gif">ript>rder="0" style="vertical-align:bottom" width="162" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0096300316305069-si6.gif">ript>r hidden">roll">row>Lprow>(riant="double-struck">R2)row>⊂r>riant="script">M.r>row>p,qrow>row>(riant="double-struck">R2)row>row> with 1 < p < q < ∞, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013).