Document Zbl 1520.35111

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Malays. Math. Sci. Soc. (2)</a> <a href="/?q=in%3A500115" title="Articles in this Issue">46, No. 5, Paper No. 166, 16 p. (2023)</a>. </div> <div class="abstract">Summary: In this study, we investigate the long-time behavior of the global solution of the anisotropic quasi-geostrophic equation, denoted by $\theta$, where $\theta$ belongs to the space $C_b ({\mathbb{R}}^+, H^s({\mathbb{R}}^2))$. Our results demonstrate that the solution decays to zero as time approaches infinity in the $L^p ({\mathbb{R}}^2)$ norm, with $p \geq 2$. Additionally, we establish that the limit of $\| \theta (t) \|_{H^s}$ approaches zero as $t$ tends to infinity.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q30" title="MSC2020">35Q30</a> </td> <td class="space"> Navier-Stokes equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q31" title="MSC2020">35Q31</a> </td> <td class="space"> Euler equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q86" title="MSC2020">35Q86</a> </td> <td class="space"> PDEs in connection with geophysics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76D05" title="MSC2020">76D05</a> </td> <td class="space"> Navier-Stokes equations for incompressible viscous fluids </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76N10" title="MSC2020">76N10</a> </td> <td class="space"> Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A86A05" title="MSC2020">86A05</a> </td> <td class="space"> Hydrology, hydrography, oceanography </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A86A10" title="MSC2020">86A10</a> </td> <td class="space"> Meteorology and atmospheric physics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76U65" title="MSC2020">76U65</a> </td> <td class="space"> Rossby waves </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B65" title="MSC2020">35B65</a> </td> <td class="space"> Smoothness and regularity of solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asurface+quasi-geostrophic+equation">surface quasi-geostrophic equation</a>; <a href="/?q=ut%3Aanisotropic+dissipation">anisotropic dissipation</a>; <a href="/?q=ut%3Aglobal+regularity">global regularity</a>; <a href="/?q=ut%3Afractional+partial+differential+equations">fractional partial differential equations</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1520.35111" data-ciurl="/ci/07729091" data-biburl="/bibtex/07729091.bib" data-amsurl="/amsrefs/07729091.bib" data-xmlurl="/xml/07729091.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07729091.pdf" title="Zbl 1520.35111 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s40840-023-01564-5" aria-label="DOI for “Long-time behavior of global solutions of anisotropic quasi-geostrophic equations in Sobolev space”" title="10.1007/s40840-023-01564-5">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2201.10840"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Aberqi, A.; Bennouna, J.; Benslimane, O.; Ragusa, MA, Existence results for double phase problem in Sobolev-Orlicz spaces with variable exponents in complete manifold, Mediterr. 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