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(2024)</a>. </div> <div class="abstract">Summary: In this article we study some problems related to the incompressible 3D Navier-Stokes equations from the point of view of Lebesgue spaces of variable exponent. These functional spaces present some particularities that make them quite different from the usual Lebesgue spaces: indeed, some of the most classical tools in analysis are not available in this framework. We will give here some ideas to overcome some of the difficulties that arise in this context in order to obtain different results related to the existence of mild solutions for this evolution problem.</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%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%3A47H10" title="MSC2020">47H10</a> </td> <td class="space"> Fixed-point theorems </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%3ALebesgue+spaces+of+variable+exponent">Lebesgue spaces of variable exponent</a>; <a href="/?q=ut%3ANavier-Stokes+equations">Navier-Stokes equations</a>; <a href="/?q=ut%3Amild+solutions">mild solutions</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1536.35240" data-ciurl="/ci/07829523" data-biburl="/bibtex/07829523.bib" data-amsurl="/amsrefs/07829523.bib" data-xmlurl="/xml/07829523.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07829523.pdf" title="Zbl 1536.35240 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/s11117-024-01043-6" aria-label="DOI for “Lebesgue spaces with variable exponent: some applications to the Navier-Stokes equations”" title="10.1007/s11117-024-01043-6">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2309.10420"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://hal.science/hal-04209551" title="Open Access at HAL">HAL </a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cannone, M.: Ondelettes, paraproduits et Navier-Stokes, PhD thesis, Paris 9, (1994)</td> </tr><tr> <td>[2]</td> <td class="space">Cannone, M.; Wu, G., Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal. Theory Methods Appl., 75, 3754-3760, 2012 · <a href="/1238.35083" class="nowrap">Zbl 1238.35083</a> · <a href="https://doi.org/10.1016/j.na.2012.01.029" class="nowrap">doi:10.1016/j.na.2012.01.029</a></td> </tr><tr> <td>[3]</td> <td class="space">Chamorro, D.: Mixed Sobolev-like inequalities in Lebesgue spaces of variable exponents and in Orlicz spaces. Positivity, 26 , Paper No. 5, 21 (2022) · <a href="/1494.46031" class="nowrap">Zbl 1494.46031</a></td> </tr><tr> <td>[4]</td> <td class="space">Cruz-Uribe, DV; Fiorenza, A., Variable Lebesgue Spaces: Foundations and Harmonic Analysis, 2013, Berlin: Springer, Berlin · <a href="/1268.46002" class="nowrap">Zbl 1268.46002</a> · <a href="https://doi.org/10.1007/978-3-0348-0548-3" class="nowrap">doi:10.1007/978-3-0348-0548-3</a></td> </tr><tr> <td>[5]</td> <td class="space">Diening, L.; Harjulehto, P.; Hästö, P.; Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, 2011, Berlin: Springer, Berlin · <a href="/1222.46002" class="nowrap">Zbl 1222.46002</a> · <a href="https://doi.org/10.1007/978-3-642-18363-8" class="nowrap">doi:10.1007/978-3-642-18363-8</a></td> </tr><tr> <td>[6]</td> <td class="space">Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269-315, 1964 · <a href="/0126.42301" class="nowrap">Zbl 0126.42301</a> · <a href="https://doi.org/10.1007/BF00276188" class="nowrap">doi:10.1007/BF00276188</a></td> </tr><tr> <td>[7]</td> <td class="space">Giga, Y.; Miyakawa, T., Navier-Stokes flow in \(\mathbb{R}^3\) with measures as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ., 14, 577-618, 1989 · <a href="/0681.35072" class="nowrap">Zbl 0681.35072</a> · <a href="https://doi.org/10.1080/03605308908820621" class="nowrap">doi:10.1080/03605308908820621</a></td> </tr><tr> <td>[8]</td> <td class="space">Grafakos, L., Classical and Modern Fourier Analysis, 2004, Hoboken: Prentice Hall, Hoboken · <a href="/1148.42001" class="nowrap">Zbl 1148.42001</a></td> </tr><tr> <td>[9]</td> <td class="space">Kato, T., Strong solutions of the Navier-Stokes equation in Morrey spaces, Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society, 22, 127-155, 1992 · <a href="/0781.35052" class="nowrap">Zbl 0781.35052</a> · <a href="https://doi.org/10.1007/BF01232939" class="nowrap">doi:10.1007/BF01232939</a></td> </tr><tr> <td>[10]</td> <td class="space">Koch, H.; Tataru, D., Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 22-35, 2001 · <a href="/0972.35084" class="nowrap">Zbl 0972.35084</a> · <a href="https://doi.org/10.1006/aima.2000.1937" class="nowrap">doi:10.1006/aima.2000.1937</a></td> </tr><tr> <td>[11]</td> <td class="space">Le Jan, Y.; Sznitman, AS, Cascades aléatoires et équations de Navier-Stokes, Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 324, 823-826, 1997 · <a href="/0876.35083" class="nowrap">Zbl 0876.35083</a></td> </tr><tr> <td>[12]</td> <td class="space">Lei, Z.; Lin, F-H, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64, 1297-1304, 2011 · <a href="/1225.35165" class="nowrap">Zbl 1225.35165</a> · <a href="https://doi.org/10.1002/cpa.20361" class="nowrap">doi:10.1002/cpa.20361</a></td> </tr><tr> <td>[13]</td> <td class="space">Lemarié-Rieusset, PG, Recent Developments in the Navier-Stokes Problem, 2002, Boca Raton: CRC Press, Boca Raton · <a href="/1034.35093" class="nowrap">Zbl 1034.35093</a> · <a href="https://doi.org/10.1201/9780367801656" class="nowrap">doi:10.1201/9780367801656</a></td> </tr><tr> <td>[14]</td> <td class="space">Lemarié-Rieusset, PG, The Navier-Stokes Problem in the 21st Century, 2018, Boca Raton: CRC Press, Boca Raton · <a href="/1342.76029" class="nowrap">Zbl 1342.76029</a> · <a href="https://doi.org/10.1201/9781315373393" class="nowrap">doi:10.1201/9781315373393</a></td> </tr><tr> <td>[15]</td> <td class="space">Taylor, ME, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Commun. Partial Differ. Equ., 17, 1407-1456, 1992 · <a href="/0771.35047" class="nowrap">Zbl 0771.35047</a> · <a href="https://doi.org/10.1080/03605309208820892" class="nowrap">doi:10.1080/03605309208820892</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. 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