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Register a new user$varepsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations
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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.
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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].
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.
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $|Pi|_{L^{p,infty}(0,T; L ^{q,infty}(mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=2$ $({3}/{2}<q<infty)$ or $| ablaPi|_{L^{p,infty}(0,T; L ^{q,infty}(mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=3$ $(1<q<infty)$ is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of $ abla u$ obtained here generalize known ones to allow the time direction to belong to Lorentz spaces.
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 $partial_3u_3$.
In this paper, we derive several new sufficient conditions of non-breakdown of strong solutions for for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant $varepsilon$ such that the solution $(rho,u,theta)$ to full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds (1) $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $uin L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$| u|_{L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=1, q>3;$$ (2) $lambda<3mu,$ $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $thetain L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$|theta|_{L^{p,infty}(0,T; L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=2, q>3/2.$$ To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces to the full Navier-Stokes system. Third, without the condition on $rho$ in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of vacuum in these systems could be allowed.
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