Pointwise descriptions of nearly incompressible vector fields with bounded curl

TL;DR

This paper characterizes nearly incompressible vector fields with bounded curl and divergence in a pointwise manner, especially focusing on those with specific growth at infinity, advancing understanding in vector calculus and fluid dynamics.

Contribution

It provides a novel pointwise characterization of nearly incompressible vector fields with bounded curl and divergence, including growth conditions at infinity.

Findings

Characterization of vector fields with bounded curl in higher dimensions.

Description of vector fields with bounded divergence and curl in two dimensions.

Extension of pointwise criteria to fields with specific growth at infinity.

Abstract

Among those nearly incompressible vector fields ${\bf{v}}:{\mathbb{R}}^n\to{\mathbb{R}}^n$ with $|x|\log|x|$ growth at infinity, we give a pointwise characterization of the ones for which $\operatorname{curl}{\bf{v}}= D{\bf{v}}-D^t{\bf{v}}$ belongs to $L^\infty$. When $n=2$ we can go further and describe, still in pointwise terms, the vector fields ${\bf{v}}:{\mathbb{R}}^2\to{\mathbb{R}}^2$ for which $|\operatorname{div}{\bf{v}}|+|\operatorname{curl}{\bf{v}}|\in L^\infty$.