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 $d \in \{2, 3\}$, ${\textbf{u}} := (u_{1}, \ldots ,u_{d})$ is the fluid velocity, $\mathbb{F}:= (F_{i,j})_{1\le i,j\le d}$ is the forcing tensor, and for all $k \in \{1, \ldots , d\}$, $\mathcal{R}_{k}$ is the $k$-th Riesz transform.