## Regularity Criteria in Lorentz Spaces for the Three Dimensional Navier-Stokes Equations

Zhou Daoguo,

School of Mathematics and Information Science, Henan Polytechnic University, Henan Jiaozuo 454000

 基金资助: 国家自然科学基金.  12071113

 Fund supported: the NSFC.  12071113

Abstract

We prove regularity criteria for weak solutions to the three dimensional Navier-Stokes equations, via horizontal part of the velocity, or the vorticity, or the gradient of velocity in scaling invariant Lorentz spaces. Our results improve almost all known regularity criteria involving Lorentz spaces or two components.

Keywords： Navier-Stokes equations ; Weak solutions ; Regularity

Zhou Daoguo. Regularity Criteria in Lorentz Spaces for the Three Dimensional Navier-Stokes Equations. Acta Mathematica Scientia[J], 2021, 41(5): 1396-1404 doi:

## 1 引言

$$$\left\{ \begin{array}{ll} &u_{t} -\Delta u+ u\cdot \nabla u +\nabla \Pi = 0, \\ & \mathrm{div}\, u = 0, \\ &u|_{t = 0} = u_0, \end{array} \right.$$$

$\bullet$ Bosia, Pata和Robinson [6], Chen和Price [9], Takahashi [14], Sohr[24], Kozono和Kim[13], Wang, Wang和Yu[17]

$$$\|u\|_{L^{p, \infty} (0, T;L^{q, \infty}( {\mathbb R}^{3}))}\leq\varepsilon, \quad 2/p+3/q = 1, \;q>3.$$$

$\bullet$ Berselli和Manfrin[5], He和Wang[23, 25], Ji, Wang和Wei[11], Wang, Wang和Yu[17]

$$$\|\nabla u\|_{L^{p, \infty} (0, T;L^{q, \infty}( {\mathbb R}^{3}))}\leq\varepsilon, \quad 2/p+3/q = 2, \;q>3/2.$$$

$\bullet$ Ji, Wang和Wei[11], Wang, Wang和Yu[17]

$$$\|\omega\|_{L^{p, \infty} (0, T;L^{q, \infty}( {\mathbb R}^{3}))}\leq\varepsilon, \quad 2/p+3/q = 2 , \;q>3/2.$$$

(1)   $u_{1}, u_{2} \in L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))$

$$$\|(u_{1}, u_{2})\|_{L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))} \leq\varepsilon_{1}, \; \; \; \; \; \; \; 2/p+3/q = 1 , \; q>3;$$$

(2)   $\nabla_{h}u_{1}, \nabla_{h}u_{2} \in L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))$

$$$\|(\nabla_{h}u_{1}, \nabla_{h}u_{2})\|_{L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))} \leq\varepsilon_{1}, \; \; \; \; \; \; \; 2/p+3/q = 2 , \; q>3/2;$$$

(3)   $\omega_{1}, \omega_{2} \in L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))$ and

$$$\|(\omega_{1}, \omega_{2})\|_{L^{p, \infty}(0, T; L ^{q, \infty}({\mathbb R}^{3}))} \leq\varepsilon_{1}, \; \; \; \; \; \; \; 2/p+3/q = 2, \; 3/2<q<\infty.$$$

$\phi $$[0, T] 上有界. 我们还需要下列初等代数事实. 引理 1.2[11] 设 (p, q) 满足 \frac{2}{p}+ \frac{3}{q} = a$$ a, p, q \geq1$. 给定$b, c_0\geq1$, $\kappa_0\in[0, 1]$, 其中, $2b+3\geq ab$,

$$$\frac{\rm d}{{\rm d}t}\int_{{\mathbb R}^{3}}|\nabla u|^{2}{\rm d}x\leq C \|\nabla_{h} u_{h}\|^{ \frac{2q_k}{2q_k-3}}_{L^{q_k, \infty}({\mathbb R}^{3})}\|\nabla u\|_{L^{2}({\mathbb R}^{3})}^{2} = C \|\nabla_{h} u_{h}\|^{p_k}_{L^{q_k, \infty}({\mathbb R}^{3})}\|\nabla u\|_{L^{2}({\mathbb R}^{3})}^{2}.$$$

$$$\|\nabla_{h} u_{h}\|^{p_{\kappa}}_{L^{q_{\kappa}, \infty}({\mathbb R}^{3})}\leq \|\nabla_{h} u_{h}\|^{p(1-\kappa)}_{L^{q , \infty}({\mathbb R}^{3})}\| \nabla_{h} u_{h} \|^{4\kappa}_{L^{2 }({\mathbb R}^{3})}\leq \|\nabla_{h} u_{h}\|^{p(1-\kappa)}_{L^{q , \infty}({\mathbb R}^{3})}\| \nabla u \|^{4\kappa}_{L^{2 }({\mathbb R}^{3})}.$$$

### 2.3 定理1.1(3) 的证明

Navier-Stokes方程(1.1) 的涡度方程为

$$$\frac12 \frac{\rm d}{{\rm d}t}\int_{{\mathbb R}^{3}}|\omega|^{2}{\rm d}x+\int_{{\mathbb R}^{3}}|\nabla \omega|^2{\rm d}x = \int_{{\mathbb R}^{3}}\omega \cdot\nabla u \cdot \omega {\rm d}x.$$$

$$$\|\nabla I_{h}\|_{L^{q_k, \infty}({\mathbb R}^{3})}\leq C\| \omega_{h} \|_{L^{q_k, \infty}({\mathbb R}^{3})}, \quad 1<q_k<\infty.$$$

$\begin{eqnarray} \int_{{\mathbb R}^{3}} (\omega\cdot\nabla u)\cdot\omega_{h} {\rm d}x &\leq &\|\omega_{h}\|_{L^{q_k, \infty}({\mathbb R}^{3})}\|\nabla u \|_{L^{ \frac{2q_k}{q_k-1}, 2}({\mathbb R}^{3})}\|\omega \|_{L^{ \frac{2q_k}{q_k-1}, 2}({\mathbb R}^{3})} {}\\ &\leq &C\|\omega_{h}\|_{L^{q_k, \infty}({\mathbb R}^{3})}\|\omega \|^{2}_{L^{ \frac{2q_k}{q_k-1}, 2}({\mathbb R}^{3})} {}\\ &\leq &C\|\omega_{h}\|_{L^{q_k, \infty}({\mathbb R}^{3})}\| \omega\|^{2- \frac{3}{q_k}}_{L^{2}({\mathbb R}^{3})}\|\nabla w\|^{ \frac{3}{q_k}}_{L^{2}({\mathbb R}^{3})}{}\\ &\leq& C \|\omega_{h}\|^{ \frac{2q_k}{2q_k-3}}_{L^{q_k, \infty}({\mathbb R}^{3})}\|\omega\|_{L^{2}({\mathbb R}^{3})}^{2} + \frac18\| \nabla \omega\|^{2}_{L^{2}({\mathbb R}^{3})}. \end{eqnarray}$

$$$\|\omega_{h}\|^{p_{\kappa}}_{L^{q_{\kappa}, \infty}({\mathbb R}^{3})}\leq \|\omega_{h}\|^{p(1-\kappa)}_{L^{q , \infty}({\mathbb R}^{3})}\| \omega_{h} \|^{4\kappa}_{L^{2 }({\mathbb R}^{3})}\leq \|\omega_{h}\|^{p(1-\kappa)}_{L^{q , \infty}({\mathbb R}^{3})}\| \omega\|^{4\kappa}_{L^{2 }({\mathbb R}^{3})}.$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Serrin J .

On the interior regularity of weak solutions of the Navier-Stokes equations

Arch Ration Mech Anal, 1962, 9, 187- 195

Bae H , Choe H .

A regularity criterion for the Navier-Stokes equations

Comm Partial Differential Equations, 2007, 32, 1173- 1187

Bae H , Wolf J .

A local regularity condition involving two velocity components of Serrin-type for the Navier-Stokes equations

C R Math Acad Sci Paris, 2016, 354, 167- 174

Beirao da Veiga H .

A new regularity class for the Navier-Stokes equations in ${\mathbb R}.{n}$

Chinese Annals of Mathematics Series B, 1995, 16, 407- 412

Berselli L , Manfrin R .

On a theorem by Sohr for the Navier-Stokes equations

J Evol Equa, 2004, 4, 193- 211

Bosia S , Pata V , Robinson J .

A weak-$L.p$ Prodi-Serrint type regularity criterion for the Navier-Stokes equations

J Math Fluid Mech, 2014, 16, 721- 725

Chae D , Choe H .

Regularity of solutions to the Navier-Stokes equation

Electron J Differential Equations, 1999, 5, 1- 7

Chen Q , Zhang Z .

Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in ${\mathbb R}.{3}$

J Differential Equations, 2005, 216, 470- 481

Chen Z , Price W .

Blow-up rate estimates for weak solutions of the Navier-Stokes equations

R Soc Lond Proc Ser A Math Phys Eng Sci, 2001, 457, 2625- 2642

Dong B , Chen Z .

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

J Math Anal Appl, 2008, 338, 1- 10

Ji X , Wang Y , Wei W .

New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier-Stokes equations

J Math Fluid Mech, 2020, 22 (1): Artcile 13

Jia X , Zhou Y .

Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of $3\times3$ mixture matrices

Nonlinearity, 2015, 28, 3289- 3307

Kim H , Kozono H .

Interior regularity criteria in weak spaces for the Navier-Stokes equations

Manuscripta Math, 2004, 115, 85- 100

Takahashi S .

On interior regularity criteria for weak solutions of the Navier-Stokes equations

Manuscripta Math, 1990, 69, 237- 254

Wang W , Zhang Z .

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

J Anal Math, 2014, 123, 139- 170

Wang W , Zhang L , Zhang Z .

On the interior regularity criteria of the 3-D Navier-Stokes equations involving two velocity components

Discrete Contin Dyn Syst, 2018, 38, 2609- 2627

Wang Y, Wei W, Yu H. $\varepsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations. 2019, arXiv: 1909.09957

Wang Y , Wu G , Zhou D .

Some interior regularity criteria involving two components for weak solutions to the 3D Navier-Stokes equations

J Math Fluid Mech, 2018, 20, 2147- 2159

Löfström J . Interpolation Spaces. Berlin: Springer-Verlag, 1976

Grafakos L . Classical Fourier Analysis. New York: Springer, 2014

Leray J .

Sur le mouvement déun liquide visqueux emplissant léspace

Acta Math, 1934, 63, 193- 248

Malý J. Advanced theory of differentiation-Lorentz spaces. 2003. http://www.karlin.mff.cuni.cz/~maly/lorentz.pdf

He C , Wang Y .

On the regularity criteria for weak solutions to the magnetohydrodynamic equations

J Differential Equations, 2007, 238, 1- 17

Sohr H .

A regularity class for the Navier-Stokes equations in Lorentz spaces

J Evol Equa, 2001, 1, 441- 467

He C , Wang Y .

Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations

Sci China Math, 2010, 53, 1767- 1774

O'Neil R .

Convolution operators and $L.{p, q}$ spaces

Duke Math J, 1963, 30, 129- 142

Tartar L .

Imbedding theorems of Sobolev spaces into Lorentz spaces

Boll Unione Mat Ital Sez B Artic Ric Mat, 1998, 1, 479- 500

Carrillo J , Ferreira L .

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

Monatsh Math, 2007, 151, 111- 142

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