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<p>The micropolar fluid system is a model based on the Navier-Stokes equations which considers two coupled variables: the velocity field <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\vec {u}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> and the microrotation field <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\vec {\omega }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>. Assuming an additional condition over the variable <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\vec {u}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> we will first prove that weak solutions <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\((\vec {u}, \vec {\omega })\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>u</mi> <mo stretchy="false">→</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of this system are smooth. Then, we will present a concentration effect of the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^3_x\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>x</mi> <mn>3</mn> </msubsup> </math></EquationSource> </InlineEquation> norm of the velocity field <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1059_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\vec {u}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> near a possible singular time. Note that some of these results can be extended to the classical Navier–Stokes equations with a suitable external force.</p>

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Partial regularity and \(L^3\)-norm concentration effects around possible blow-up points for the micropolar fluid equations

  • 期刊论文

  • 发表日期：

    2025年5月10日

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  • Diego Chamorro,
  • David Llerena

摘要

The micropolar fluid system is a model based on the Navier-Stokes equations which considers two coupled variables: the velocity field \(\vec {u}\) u → and the microrotation field \(\vec {\omega }\) ω → . Assuming an additional condition over the variable \(\vec {u}\) u → we will first prove that weak solutions \((\vec {u}, \vec {\omega })\) ( u → , ω → ) of this system are smooth. Then, we will present a concentration effect of the \(L^3_x\) L x 3 norm of the velocity field \(\vec {u}\) u → near a possible singular time. Note that some of these results can be extended to the classical Navier–Stokes equations with a suitable external force.