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<p>In this paper, we present some new regularity criteria for suitable weak solutions to the 3D simplified Ericksen-Leslie system in critical spaces. Namely, it is shown that suitable weak solutions to this system are regular if the velocity field belongs to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_776_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="171" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty }(-1,0;BMO^{-1}(\mathbb {R}^3))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>;</mo> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or the velocity field and direct field are in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_776_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="151" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty }(-1,0;\dot{B}^{-1}_{\infty ,\infty }(\mathbb {R}^3))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>;</mo> <msubsup> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> <mrow> <mi>∞</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our results improve previous corresponding criteria due to Liu, Min and Zhang [J. Differential Equations <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_776_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{267}\)</EquationSource> <EquationSource Format="MATHML"><math> <mn mathvariant="bold">267</mn> </math></EquationSource> </InlineEquation>, 2643–2670 (2019)] and Men, Wang and Wu [Math. Methods Appl. Sci. <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_776_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{41}\)</EquationSource> <EquationSource Format="MATHML"><math> <mn mathvariant="bold">41</mn> </math></EquationSource> </InlineEquation>, 3672–3683 (2018)].</p>

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On the regularity for the simplified Ericksen-Leslie system in critical spaces

  • 期刊论文

  • 发表日期：

    2025年3月25日

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  • Zhongbao Zuo

摘要

In this paper, we present some new regularity criteria for suitable weak solutions to the 3D simplified Ericksen-Leslie system in critical spaces. Namely, it is shown that suitable weak solutions to this system are regular if the velocity field belongs to \(L^{\infty }(-1,0;BMO^{-1}(\mathbb {R}^3))\) L ∞ ( - 1 , 0 ; B M O - 1 ( R 3 ) ) or the velocity field and direct field are in \(L^{\infty }(-1,0;\dot{B}^{-1}_{\infty ,\infty }(\mathbb {R}^3))\) L ∞ ( - 1 , 0 ; B ˙ ∞ , ∞ - 1 ( R 3 ) ) . Our results improve previous corresponding criteria due to Liu, Min and Zhang [J. Differential Equations \(\textbf{267}\) 267 , 2643–2670 (2019)] and Men, Wang and Wu [Math. Methods Appl. Sci. \(\textbf{41}\) 41 , 3672–3683 (2018)].