Document Zbl 1346.76011

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Ration. Mech. Anal.</a> <a href="/?q=in%3A262302" title="Articles in this Issue">197, No. 1, 297-336 (2010)</a>. </div> <div class="abstract">Summary: This paper is concerned with a simplified hydrodynamic equation, proposed by <span class="zbmathjax-textit">J. L. Ericksen</span> [Arch. Ration. Mech. Anal. 9, 371–378 (1962; <a href="/0105.23403">Zbl 0105.23403</a>)] and <span class="zbmathjax-textit">F. M. Leslie</span> [ibid. 28, 265–283 (1968; <a href="/0159.57101">Zbl 0159.57101</a>)], modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and boundary regularity theorems for such a flow under smallness conditions. As a consequence, we establish the existence of global (in time) weak solutions on a bounded smooth domain in ${\mathbb{R}^2}$ which are smooth everywhere with possible exceptions of finitely many singular times.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5782722">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5782722">Cited in <strong>234</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76A15" title="MSC2020">76A15</a> </td> <td class="space"> Liquid crystals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82D30" title="MSC2020">82D30</a> </td> <td class="space"> Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q35" title="MSC2020">35Q35</a> </td> <td class="space"> PDEs in connection with fluid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anematic+liquid+crystals">nematic liquid crystals</a>; <a href="/?q=ut%3Aregularity">regularity</a>; <a href="/?q=ut%3Aexistence">existence</a>; <a href="/?q=ut%3Aweak+solution">weak solution</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0105.23403">Zbl 0105.23403</a>; <a href="/0159.57101">Zbl 0159.57101</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 1346.76011" data-ciurl="/ci/05782722" data-biburl="/bibtex/05782722.bib" data-amsurl="/amsrefs/05782722.bib" data-xmlurl="/xml/05782722.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05782722.pdf" title="Zbl 1346.76011 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/s00205-009-0278-x" aria-label="DOI for “Liquid crystal flows in two dimensions”" title="10.1007/s00205-009-0278-x">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Chang K.C.: Heat flow and boundary value problem for harmonic maps. 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CPAM XLVIII, 501–537 (1995) · <a href="/0842.35084" class="nowrap">Zbl 0842.35084</a></td> </tr><tr> <td>[15]</td> <td class="space">Lin F.H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. DCDS 2(1), 1–22 (1998) · <a href="/0948.35098" class="nowrap">Zbl 0948.35098</a></td> </tr><tr> <td>[16]</td> <td class="space">Lin F.H., Wang C.Y.: Harmonic and quasi-harmonic spheres. II. Comm. Anal. Geom. 10(2), 341–375 (2002) · <a href="/1042.58005" class="nowrap">Zbl 1042.58005</a></td> </tr><tr> <td>[17]</td> <td class="space">Qing J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3(1–2), 297–315 (1995) · <a href="/0868.58021" class="nowrap">Zbl 0868.58021</a></td> </tr><tr> <td>[18]</td> <td class="space">Seregin G., Shilkin T., Solonnikov V.: Boundary partial regularity for the Navier-Stokes equations. J. Math. 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(2) 113(1), 1–24 (1981) · <a href="/0462.58014" class="nowrap">Zbl 0462.58014</a></td> </tr><tr> <td>[22]</td> <td class="space">Schoen, R., Uhlenbeck, K.: Approximation of Sobolev maps between Riemannian manifolds. Preprint (1984) · <a href="/0555.58011" class="nowrap">Zbl 0555.58011</a></td> </tr><tr> <td>[23]</td> <td class="space">Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici 60, 558–581 (1985) · <a href="/0595.58013" class="nowrap">Zbl 0595.58013</a> · <a href="https://doi.org/10.1007/BF02567432" class="nowrap">doi:10.1007/BF02567432</a></td> </tr><tr> <td>[24]</td> <td class="space">Temam, R.: Navier-Stokes equations. Studies in Mathematics and its Applications, Vol. 2, North Holland, Amsterdam, 1977 · <a href="/0383.35057" class="nowrap">Zbl 0383.35057</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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