论文标题
各向异性Lebesgue空间中速度的一个定向导数的规律性标准到3D Navier-Stokes方程
Regularity criteria via one directional derivative of the velocity in anisotropic Lebesgue spaces to the 3D Navier-Stokes equations
论文作者
论文摘要
在本文中，我们考虑了3D不可压缩的Navier-Stokes方程的规律性标准，该方程是各向异性Lebesgue空间中速度的一个定向导数。更准确地说，如果$ \ partial_3u $满足$ \ int^{t} _ {0} \ frac {\ left \ | \ | \ | \ | \ | \ | \ | \ | \ | \ partial_3 u（t） \ right \ | _ {l^q_ {x_2}} \ right \ |^β_{l^{l^{r} _ {x_3}}} {1 + \ ln \ left（\ | \ | \ | \ | \ partial_3u $ \ text {where} \ frac {2}β+\ \ \ frac {1} {p}+\ frac {1} {q} {q}+\ frac {1} {r} {r} = 1 \ text {and} 1- \ left（\ frac {1} {p}+\ frac {1} {q}+\ frac {1} {r} {r} \ right）\ geq 0 $。
In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if the $\partial_3u$ satisfies $$\int^{T}_{0} \frac{\left\|\left\|\left\|\partial_3 u(t) \right\|_{L^p_{x_1}} \right\|_{L^q_{x_2}} \right\|^β_{L^{r}_{x_3}}} {1 + \ln\left(\|\partial_3u \left(t\right)\|_{L^2} + e\right)}dt < \infty,$$ $\text { where } \frac{2}β+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 \text { and } 2 < p, q, r \leq \infty, 1-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right) \geq 0 $.
