On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space

Abstract

We give some conditions under which the pressure term in the incompressible Navier-Stokes equations on the entire $d$-dimensional Euclidean space is determined by the formula $\nabla p=\nabla \left(\sum _{i,j=1}^d{\mathcal{R}}_i{\mathcal{R}}_j(u_i{u}_j-F_{i,j})\right)$, where $d\in \{2,3\}$, $\text{u}:=(u_1,\dots ,u_d)$ is the fluid velocity, $\mathbb{F}:=(F_{i,j}{)}_{1\le i,j\le d}$ is the forcing tensor, and for all $k\in \{1,\dots ,d\}$, ${\mathcal{R}}_k$ is the $k$-th Riesz transform.

12 pages