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 id="mmlsi1" class="mathmlsrc">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&isin;L²(R²)iner hidden">i1.gif" overflow="scroll">row>i>θi>0&isin;i>Li>2row>(i mathvariant="double-struck">Ri>2)row>row>. Let θ   and id="mmlsi2" class="mathmlsrc">itle="View the MathML source" 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"><img class="imgLazyJSB inlineImage" height="18" width="13" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si2.gif">ript><img height="18" border="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>iner hidden">i2.gif" overflow="scroll">r accent="true">i>θi>&tilde;r> be two weak solutions with the same initial value θ₀. If θ satisfies the energy inequality and ifiv class="formula" id="ueq0001">iv class="mathml">id="mmlsi3" class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305069&_mathId=si3.gif&_user=111111111&_pii=S0096300316305069&_rdoc=1&_issn=00963003&md5=bcb86f51bfde77b62568a3d40d902dd7"><img class="imgLazyJSB smallImg inlineImage" height="44" width="607" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si3.gif">ript><img height="44" border="0" style="vertical-align:bottom" width="607" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0096300316305069-si3.gif">ript>iner hidden">i3.gif" overflow="scroll">row>i>∇i>i>θi>&isin;i>Li>i>ri>0,i>Ti>;r>i mathvariant="script">Mi>.r>row>i>pi>,i>qi>row>row>(i mathvariant="double-struck">Ri>2)row>idth="4.pt">withidth="4.pt">idth="4.pt">rac>2i>pi>rac>+rac>i>αi>i>ri>rac>=i>αi>+i>βi>&minus;1idth="4.pt">andidth="4.pt">rac>2row>i>αi>+i>βi>&minus;1row>rac><i>pi><rac>3i>ri>rac>,row><img class="temp" src="/sd/blank.gif">iv>iv>where id="mmlsi4" class="mathmlsrc">itle="View the MathML source" 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"><img class="imgLazyJSB inlineImage" height="25" width="79" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si4.gif">ript><img height="25" border="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>iner hidden">i4.gif" overflow="scroll">row>r>i mathvariant="script">Mi>.r>row>i>pi>,i>qi>row>row>(i mathvariant="double-struck">Ri>2)row>row> denotes the homogeneous Morrey–Campanato space, then we have id="mmlsi5" class="mathmlsrc">itle="View the MathML source" 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"><img class="imgLazyJSB inlineImage" height="18" width="49" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si5.gif">ript><img height="18" border="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>iner hidden">i5.gif" overflow="scroll">row>i>θi>=r accent="true">i>θi>&tilde;r>row>. Due to the embedding relation id="mmlsi6" class="mathmlsrc">itle="View the MathML source" 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"><img class="imgLazyJSB inlineImage" height="25" width="162" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300316305069-si6.gif">ript><img height="25" border="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>iner hidden">i6.gif" overflow="scroll">row>i>Li>i>pi>row>(i mathvariant="double-struck">Ri>2)row>⊂r>i mathvariant="script">Mi>.r>row>i>pi>,i>qi>row>row>(i mathvariant="double-struck">Ri>2)row>row> with 1 < p < q < &infin;, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013).