The Jones Strong Distribution Banach Spaces

TL;DR

This paper introduces a new class of Banach spaces called Jones strong distribution spaces that embed classical function spaces and test functions, with unique invariance properties under derivatives, and applies them to derive bounds for Navier-Stokes equations.

Contribution

The paper constructs the Jones strong distribution Banach spaces, demonstrating their embedding properties and invariance under derivatives, and applies these spaces to fluid dynamics problems.

Findings

Spaces contain $L^p$ and test functions as dense embeddings

Invariance of norm under distributional derivatives

New a priori bounds for Navier-Stokes equations

Abstract

In this note, we introduce a new class of separable Banach spaces, ${SD^p}[{\mathbb{R}^n}],\;1 \leqslant p \leqslant \infty$, which contain each $L^p$-space as a dense continuous and compact embedding. They also contain the nonabsolutely integrable functions and the space of test functions ${\mathcal{D}}[{\mathbb{R}^n}]$, as dense continuous embeddings. These spaces have the remarkable property that, for any multi-index $\alpha, \; \left\| {{D^\alpha }{\mathbf{u}}} \right\|_{SD} = \left\| {\mathbf{u}} \right\|_{SD}$, where $D$ is the distributional derivative. We call them Jones strong distribution Banach spaces because of the crucial role played by two special functions introduced in his book (see \cite{J}, page 249). After constructing the spaces, we discuss their basic properties and their relationship to ${\mathcal{D}}[{\mathbb{R}^n}]$ and ${\mathcal{D'}}[{\mathbb{R}^n}]$. As an application, we obtain new a priori bounds for the Navier-Stokes equation.