<p>We consider the axisymmetric Navier-Stokes equations in a finite cylinder <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$v_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$v_\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\omega _\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> vanish on the lateral part of boundary <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> of the cylinder, and that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$v_z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\omega _\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\partial _zv_\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>z</mi> </msub> <msub> <mi>v</mi> <mi>φ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> vanish on the top and bottom parts of the boundary <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, where we used standard cylindrical coordinates, and we denoted by <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\omega =\operatorname {curl}v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>=</mo> <mo>curl</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation> the vorticity field. We use <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$H^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Finally, using the estimate <Equation ID="Equ154"> <EquationSource Format="TEX">$ \Vert v_\varphi \Vert _{L_q(0,T;L_p(\Omega ))} \le A, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>v</mi> <mi>φ</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msub> <mi>L</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≤</mo> <mi>A</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <i>A</i> is a given number and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\frac{3}{p} + \frac{2}{q} < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>3</mn> <mi>p</mi> </mfrac> <mo>+</mo> <mfrac> <mn>2</mn> <mi>q</mi> </mfrac> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> we prove the existence of global regular axially-symmetric solutions.</p>

A Regularity Criterion for the Angular Component of Velocity in the norm $L_q(0,T;L_p(\Omega )),\;\frac{3}{p} +\frac{2}{q}<1,\;q< \infty $, in Axisymmetric Navier-Stokes Equations in a Cylinder

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2026年6月16日

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Wiesław J. Grygierzec,
Wojciech M. Zaja̧czkowski

摘要

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ . We assume that $v_r$ $v_{r}$ , $v_\varphi $ $v_{φ}$ , $\omega _\varphi $ $ω_{φ}$ vanish on the lateral part of boundary $\partial \Omega $ $\partial Ω$ of the cylinder, and that $v_z$ $v_{z}$ , $\omega _\varphi $ $ω_{φ}$ , $\partial _zv_\varphi $ $\partial_{z} v_{φ}$ vanish on the top and bottom parts of the boundary $\partial \Omega $ $\partial Ω$ , where we used standard cylindrical coordinates, and we denoted by $\omega =\operatorname {curl}v$ $ω = curl v$ the vorticity field. We use $H^3$ $H^{3}$ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Finally, using the estimate $ \Vert v_\varphi \Vert _{L_q(0,T;L_p(\Omega ))} \le A, $ $‖ v_{φ} ‖_{L_{q} (0, T; L_{p} (Ω))} \leq A,$ where A is a given number and $\frac{3}{p} + \frac{2}{q} < 1$ $\frac{3}{p} + \frac{2}{q} < 1$ , $q<\infty $ $q < \infty$ we prove the existence of global regular axially-symmetric solutions.

A Regularity Criterion for the Angular Component of Velocity in the norm \(L_q(0,T;L_p(\Omega )),\;\frac{3}{p} +\frac{2}{q}<1,\;q< \infty \), in Axisymmetric Navier-Stokes Equations in a Cylinder

摘要