Document Zbl 1433.35262

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Ration. Mech. Anal.</a> <a href="/?q=in%3A447720" title="Articles in this Issue">236, No. 2, 839-891 (2020)</a>. </div> <div class="abstract">Summary: In this paper, we study the Cauchy problem of the Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities – blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are Hölder continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation – the flux density remains bounded while its two components approach infinity at formations of cusp singularities.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7178295">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7178295">Cited in <strong>14</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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><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%3A35B65" title="MSC2020">35B65</a> </td> <td class="space"> Smoothness and regularity of solutions to PDEs </td> </tr></table> </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 1433.35262" data-ciurl="/ci/07178295" data-biburl="/bibtex/07178295.bib" data-amsurl="/amsrefs/07178295.bib" data-xmlurl="/xml/07178295.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07178295.pdf" title="Zbl 1433.35262 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/s00205-019-01484-4" aria-label="DOI for “Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model”" title="10.1007/s00205-019-01484-4">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1906.09658"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">Bressan, A.; Chen, G., Lipschitz metric for a class of nonlinear wave equations, Arch. 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