On the Use of the Riesz Transforms to Determine the Pressure Term in the Incompressible Navier-Stokes Equations on the Whole Space

We give some conditions under which the pressure term in incompressible Navier-Stokes equations on entire $d$ -dimensional Euclidean space is determined by 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 $d \in \{2, 3\}$ ${\textbf{u}} := (u_{1}, \ldots ,u_{d})$ fluid velocity, $\mathbb{F}:= (F_{i,j})_{1\le i,j\le d}$ forcing tensor, and for all $k \{1, d\}$ $\mathcal{R}_{k}$ $k$ -th Riesz transform.