Dynamic Refinement of Pressure Decomposition in Navier-Stokes Equations

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.