<p>In this note we consider some regularity criteria for 3D shear thickening fluids via one velocity component or two velocity components. Especially, it is proved that a weak solution for the 3D shear thickening flows becomes a strong solution, for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2031_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$2< p<\frac{11}{5}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mn>11</mn> <mn>5</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, provided that one velocity component <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2031_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">$u^3\in L^\infty (0,T;L^\alpha (\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>L</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <Equation ID="Equ67"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2031_Article_Equ67.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="225" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \alpha \ge \max \left\{ \frac{3p}{2p-2},\frac{10p}{3(5p-8)}\right\} , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>α</mi> <mo>≥</mo> <mo movablelimits="true">max</mo> <mfenced close="}" open="{"> <mfrac> <mrow> <mn>3</mn> <mi>p</mi> </mrow> <mrow> <mn>2</mn> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mn>10</mn> <mi>p</mi> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>5</mn> <mi>p</mi> <mo>-</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which is consistent with the case of Navier–Stokes equations as Cao-Titi [<CitationRef CitationID="CR7">7</CitationRef>] or Pokorný and Zhou [<CitationRef CitationID="CR22">22</CitationRef>]. For the regularity criteria depending on two velocity components <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2031_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$u_h$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation>, we also obtained some almost Serrin conditions.</p>

Regularity Criterion for a Type of 3D Shear Thickening Fluids via One Velocity Component

期刊论文
发表日期：

2025年5月24日

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Wendong Wang,
Guoxu Yang,
Jianbo Yu

摘要

In this note we consider some regularity criteria for 3D shear thickening fluids via one velocity component or two velocity components. Especially, it is proved that a weak solution for the 3D shear thickening flows becomes a strong solution, for $2< p<\frac{11}{5}$ $2 < p < \frac{11}{5}$ , provided that one velocity component $u^3\in L^\infty (0,T;L^\alpha (\Omega ))$ $u^{3} \in L^{\infty} (0, T; L^{α} (Ω))$ with $\begin{aligned} \alpha \ge \max \left\{ \frac{3p}{2p-2},\frac{10p}{3(5p-8)}\right\} , \end{aligned}$ $\begin{matrix} α \geq max \{\frac{3 p}{2 p - 2}, \frac{10 p}{3 (5 p - 8)}\}, \end{matrix}$ which is consistent with the case of Navier–Stokes equations as Cao-Titi [7] or Pokorný and Zhou [22]. For the regularity criteria depending on two velocity components $u_h$ $u_{h}$ , we also obtained some almost Serrin conditions.