论文标题
弱解的局部规律性较弱的解决方案
Local regularity of weak solutions of the hypodissipative Navier-Stokes equations
论文作者
论文摘要
我们考虑3D不可压缩性的Navier-Stokes方程，当耗散作为分数Laplacian $（-Δ）^s $（$ s \ in（\ frac34,1）$）时，我们提供了一种新的引导程序，可以在空间时间内分析弱解决方案。这包括我们本地化在空间中的几种均匀的Kato-Ponce类型换向器估计值，并且似乎适用于其他具有分数耗散的抛物线系统。我们还提供了有关压力的新估计，$ \ |（-Δ）^s p \ | _ {\ Mathcal {h}^1} \ sillsim \ | （-δ）^{\ frac S2} u \ |^2_ {l^2} $。我们应用我们的主要结果来证明，任何合适的弱解决方案$ u $都满足$ \ nabla^n u \ in l^{p，\ infty} _ {\ Mathrm {loc {loc}}（\ MathBB {r}^3 \ times（0，\ infty）$ p = \ frac frac^s $ $ n = 1,2 $。作为我们本地规律定理的必然学，我们改善了Tang-Yu的部分规律性结果[Comm。数学。 Phys。，334（30），2015年，第1455---1482页]，并获得对单数集合$ s $，$ d_b（S \ cap \ {
We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-Δ)^s$ for $s\in (\frac34,1)$, and we provide a new bootstrapping scheme that makes it possible to analyse weak solutions locally in space-time. This includes several homogeneous Kato-Ponce type commutator estimates which we localize in space, and which seems applicable to other parabolic systems with fractional dissipation. We also provide a new estimate on the pressure, $\|(-Δ)^s p \|_{\mathcal{H}^1}\lesssim \| (-Δ)^{\frac s2} u \|^2_{L^2}$. We apply our main result to prove that any suitable weak solution $u$ satisfies $\nabla^n u \in L^{p,\infty }_{\mathrm{loc}}(\mathbb{R}^3\times(0,\infty))$ for $p=\frac{2(3s-1)}{n+2s-1}$, $n=1,2$. As a corollary of our local regularity theorem, we improve the partial regularity result of Tang-Yu [Comm. Math. Phys., 334(30), 2015, pp. 1455--1482], and obtain an estimate on the box-counting dimension of the singular set $S$, $d_B(S\cap \{t\geq t_0 \} )\leq \frac13 (15-2s-8s^2) $ for every $t_0>0$.
