Document Zbl 1530.35188

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We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.</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%3A76D07" title="MSC2020">76D07</a> </td> <td class="space"> Stokes and related (Oseen, etc.) flows </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%3A76D03" title="MSC2020">76D03</a> </td> <td class="space"> Existence, uniqueness, and regularity theory for incompressible viscous fluids </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74F10" title="MSC2020">74F10</a> </td> <td class="space"> Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74M15" title="MSC2020">74M15</a> </td> <td class="space"> Contact in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35M30" title="MSC2020">35M30</a> </td> <td class="space"> Mixed-type systems of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35A01" title="MSC2020">35A01</a> </td> <td class="space"> Existence problems for PDEs: global existence, local existence, non-existence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35A02" title="MSC2020">35A02</a> </td> <td class="space"> Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afluid-solid+systems">fluid-solid systems</a>; <a href="/?q=ut%3AStokes+equations">Stokes equations</a>; <a href="/?q=ut%3Acontact+issue">contact issue</a>; <a href="/?q=ut%3Aglobal-in-time+existence+of+solutions">global-in-time existence of solutions</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 1530.35188" data-ciurl="/ci/07702835" data-biburl="/bibtex/07702835.bib" data-amsurl="/amsrefs/07702835.bib" data-xmlurl="/xml/07702835.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07702835.pdf" title="Zbl 1530.35188 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/10.3233/ASY-221790" aria-label="DOI for “Global solutions to coupled (Navier-)Stokes Newton systems in $\mathbb{R}^3$”" title="10.3233/ASY-221790">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">C. 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Kelaï, Justification of lubrication approximation: An application to fluid/solid interactions, Asymptot. Anal. 95(3-4) (2015), 187-241. · <a href="/1333.35195" class="nowrap">Zbl 1333.35195</a></td> </tr><tr> <td>[12]</td> <td class="space">M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal. 40(6) (2009), 2451-2477. doi:10.1137/080716074. · <a href="/1178.35291" class="nowrap">Zbl 1178.35291</a> · <a href="https://doi.org/10.1137/080716074" class="nowrap">doi:10.1137/080716074</a></td> </tr><tr> <td>[13]</td> <td class="space">M. Hillairet and T. Takahashi, Blow up and grazing collision in viscous fluid solid interaction systems, Ann. Inst. H. Poincaré Anal. 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Sabbagh, On the motion of several disks in an unbounded viscous incompressible fluid, Nonlinearity 32(6) (2019), 2157-2181. doi:10.1088/1361-6544/ab006f. · <a href="/1420.35402" class="nowrap">Zbl 1420.35402</a> · <a href="https://doi.org/10.1088/1361-6544/ab006f" class="nowrap">doi:10.1088/1361-6544/ab006f</a></td> </tr><tr> <td>[19]</td> <td class="space">L. Sabbagh, Etude du mouvement de solides rigides dans un fluide visqueux, 2018, Thèse de l’Université de Montpellier.</td> </tr><tr> <td>[20]</td> <td class="space">J.A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002), 113-147. doi:10.1007/s002050100172. · <a href="/1018.76012" class="nowrap">Zbl 1018.76012</a> · <a href="https://doi.org/10.1007/s002050100172" class="nowrap">doi:10.1007/s002050100172</a></td> </tr><tr> <td>[21]</td> <td class="space">T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003), 1499-1532. · <a href="/1101.35356" class="nowrap">Zbl 1101.35356</a></td> </tr><tr> <td>[22]</td> <td class="space">T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech. 6 (2004), 53-77. doi:10.1007/s00021-003-0083-4. · <a href="/1054.35061" class="nowrap">Zbl 1054.35061</a> · <a href="https://doi.org/10.1007/s00021-003-0083-4" class="nowrap">doi:10.1007/s00021-003-0083-4</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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