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

Borys Álvarez-Samaniego, Wilson P. Álvarez-Samaniego, Pedro G. Fernández-Dalgo

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 $\displaystyle \nabla p = \nabla \left(\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j - F_{i,j}) \right)$, 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.

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 -dimensional Euclidean space is determined by the formula , where , is the fluid velocity, is the forcing tensor, and for all , is the -th Riesz transform.

Paper Structure

This paper contains 2 sections, 3 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Determination of the pressure term in the incompressible Navier-Stokes equations on the full Euclidean space

Key Result

Lemma 1.1

Let $d \in \mathbb{Z}^+$, $0<\delta<d$, and $1<p<+\infty$. Then, $w_\delta$ belongs to the Muckenhoupt $\mathcal{A}_p(\mathbb{R}^d)$ class.

Theorems & Definitions (7)

  • Definition 1.1
  • Lemma 1.1: Muckenhoupt weights
  • proof
  • Corollary 1.1
  • proof
  • Theorem 2.1
  • proof