Dynamic Refinement of Pressure Decomposition in Navier-Stokes Equations

TL;DR

This paper introduces a dynamic refinement method for pressure decomposition in Navier-Stokes equations, establishing new regularity results and energy control near singularities in fluid flow models.

Contribution

It develops a novel dynamic pressure decomposition technique that improves understanding of energy behavior and regularity near singularities in Navier-Stokes solutions.

Findings

Critical energy near singularities is controlled by boundary behavior.

Refined regularity results are established for Navier-Stokes solutions.

Pressure decomposition is dynamically refined to improve analysis near paraboloid vertices.

Abstract

In this work, the local decomposition of pressure in the Navier-Stokes equations is dynamically refined to prove that a relevant critical energy of a suitable Leray-type solution inside a backward paraboloid -- regardless of its aperture -- is controlled near the vertex by a critical behavior confined to a neighborhood of the paraboloid's boundary. This neighborhood excludes the interior near the vertex and remains separated from the temporal profile of the vertex, except at the vertex itself. Then, we present a refined scaling invariant regularity result.