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Register a new userRegularity criterion for 3D Navier-Stokes Equations in Besov spaces
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Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $ abla_{h}{u}$ (or $ abla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; dot{B}_{p,r}^{s}(mathbb{R}^{3}))$, where $ abla_{h}=(partial_{x_{1}},partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $partial_3u_3$.
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In this article, we establish sufficient conditions for the regularity of solutions of Navier-Stokes equations based on one of the nine entries of the gradient tensor. We improve the recently results of C.S. Cao, E.S. Titi (Arch. Rational Mech.Anal. 202 (2011) 919-932) and Y. Zhou, M. Pokorn$acute{y}$ (Nonlinearity 23, 1097-1107 (2010)).
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Lerays blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,infty}(-1,0;L^{q,l}(mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<infty$ and $qleq l <infty$ is finite.
Several types of new regularity criteria for Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. Some of them are based on the third component $u_3$ of velocity under Prodi-Serrin index condition, another type is in terms of $omega_3$ and $partial_3u_3$ with Prodi-Serrin index condition. And a very recent work of the authors, based on only one of the nine entries of the gradient tensor, is renovated.
We establish several boundary $varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the Dirichlet boundary condition on the flat boundary. Our proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur [18], Choi-Vasseur [2], and Phuc-Guevara [6].
We define the Ladyzhenskaya-Lions exponent $alpha_{rm {tiny sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Delta)^{alpha}$ in ${Bbb R}^n$, for all $ngeq 2$. We review the proof of strong global solvability when $alphageq alpha_{rm {tiny sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1over 2} | abla u |^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $alpha<alpha_{rm {tiny sc l}} (n)$. The energy is critical under scale transformations only for $alpha=alpha_{rm {tiny sc l}} (n)$.
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