Local regularity of weak solutions of the hypodissipative Navier-Stokes equations

Abstract

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-\Delta {)}^s$ for $s\in (\frac{3}{4},1)$, 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, $\Vert (-\Delta {)}^{s}p{\Vert }_{{\mathcal{H}}^1}\lesssim \Vert (-\Delta {)}^{\frac{s}{2}}u{\Vert }_{L^2}^2$. We apply our main result to prove that any suitable weak solution $u$ satisfies ${\nabla }^{n}u\in L_{\mathrm{loc}}^{p,\infty }({\mathbb{R}}^3\times (0,\infty ))$ 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 $S$, $d_B(S\cap \{t\ge t_0\})\le \frac{1}{3}(15-2s-8s^2)$ for every $t_0>0$.

57 pages, 1 figure

References in corpus (3)

• The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
• Improved bounds for box dimensions of potential singular points to the Navier--Stokes equations
• Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation

Cited by in corpus (2)

• Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation
• Phase transitions in the fractional three-dimensional Navier-Stokes equations