New Regularity Criteria for the Navier-Stokes Equations in Terms of Pressure

Abstract

In this paper, we generalize the main results of [1] and [31] to Lorentz spaces, using a simple procedure. The main results are the following. Let $n\geq 3$ and let $u$ be a Leray-Hopf solution to the $n$-dimensional Navier-Stokes equations with viscosity $\nu$ and divergence free initial condition $u_0\in L^2(\mathbb{R}^n)\cap L^{k}(\mathbb{R}^n)$ (where $k=k(s)$ is sufficiently large). Then there exists a constant $c>0$ such that if \begin{equation} \|p\|_{L^{r,\infty}(0,\infty;L^{s,\infty}(\mathbb{R}^n))}<c\hspace{10mm}\frac{n}{s}+\frac{2}{r}\leq 2,\hspace{5mm}s>\frac{n}{2} \end{equation} or \begin{equation} \|\nabla p\|_{L^{r,\infty}(0,\infty;L^{s,\infty}(\mathbb{R}^n))}<c\hspace{10mm}\frac{n}{s}+\frac{2}{r}\leq 3,\hspace{5mm}s>\frac{n}{3} \end{equation} then $u$ is smooth on $(0, \infty) \times \mathbb{R}^n$. Partial results in the case $n=3$ were obtained in [32], [33] and then recently extended to all appropriate pairs of $r,s$ in [14]. Our results present a unified proof which works for all dimensions $n\geq 3$ and the full range or admissible pairs, $(s,r)$.