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Register a new userOn $W^{1,p}$-regularity estimate for a class of generalized Stokes systems and its applications to the Navier-Stokes equations
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Tuoc Phan
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This paper establishes global weighted Calderon-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stokes operator consist of symmetric and skew-symmetric parts, which are both discontinuous. Moreover, the skew-symmetric part is not required to be bounded and therefore it could be singular. Sufficient conditions on the coefficients are provided to ensure the global weighted $W^{1,p}$-regularity estimates for weak solutions of the systems. As a direct application, we show that our $W^{1,p}$-regularity results give some criteria in critical spaces for the global regularity of weak Leray-Hopf solutions of the Navier-Stokes system of equation
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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 consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<infty$ and $lim_{tto T}sqrt{T-t}||u(t)||_{L^infty}=infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $uin L^{2,infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,infty}([0,T];BMO(Omega))}<infty$ for arbitrary $Omegasubseteqmathbb{R}^3$ then the local energy equality is valid on $[0,T]timesOmega$. As a corollary, we also prove $||u||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}<infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)||_{BMO}$ must blow up at a rate faster than $frac{c}{sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}=M<infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}leq c_M$ then $u$ is regular on $(0,T]timesmathbb{R}^3$.
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 consider suitable weak solutions of the incompressible Navier--Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.
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].
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