Partially regular weak solutions of the Navier-Stokes equations in $\mathbb{R}^4 \times [0,\infty[$

TL;DR

This paper establishes the existence of partially regular weak solutions to the Navier-Stokes equations in four-dimensional space, with controlled singularities, using advanced compactness and measure techniques.

Contribution

It introduces a novel partial regularity result for Navier-Stokes in 4D, employing defect measures and concentration-compactness methods to handle non-compactness issues.

Findings

Existence of weak solutions with locally finite 2D parabolic Hausdorff measure of singular set.

Application of parabolic concentration-compactness theorem in 4D setting.

Use of defect measures to overcome lack of compactness in higher dimensions.

Abstract

We show that for any given solenoidal initial data in $L^2$ and any solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$, there exist partially regular weak solutions of the Navier-Stokes equations in $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular sets have locally finite $2$-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to overcome the possible lack of compactness arising in the spatially $4$-dimensional setting by using defect measures, which we then incorporate into the partial regularity theory.