A Helmholtz-type decomposition for the space of symmetric matrices

TL;DR

This paper develops a Helmholtz-type decomposition for symmetric matrix functions, enhancing understanding of the strain constraint space relevant to Navier--Stokes regularity, with geometric insights into eigenvalue distributions.

Contribution

It introduces a novel Helmholtz-type decomposition for symmetric matrices and characterizes the orthogonal complement of the strain constraint space.

Findings

Characterization of the orthogonal complement of the strain constraint space

Analysis of eigenvalue distribution geometry within the space

Enhanced understanding of the strain constraint space's structure

Abstract

In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better understanding of the strain constraint space, which is important to the Navier--Stokes regularity problem. In particular, we give a full characterization the orthogonal complement of the strain constraint space and investigate the geometry of the eigenvalue distribution of matrices in the strain constraint space.