اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBoundary partial regularity for the high dimensional Navier-Stokes equations
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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.
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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 delica
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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 $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<in
fty$ 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 $(0,T]timesmathbb{R}^3$.
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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
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