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Dynamic Refinement of Pressure Decomposition in Navier-Stokes Equations

Pedro Gabriel Fernández Dalgo

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.

Dynamic Refinement of Pressure Decomposition in Navier-Stokes Equations

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.

Paper Structure

This paper contains 18 sections, 3 theorems, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Local energy in the interior of paraboloids
  4. Boundedness of the right-hand side terms in the energy balance
  5. The $p_1$ part.
  6. The $p_2$ part.
  7. The $p_3$ part.
  8. Boundedness of the Scale-Invariant Energy
  9. Gaining Damping with the Gradient Part
  10. Poincaré inequality with the new localization
  11. Gronwall's type estimate
  12. Boundedness of g_gamma and f
  13. Boundedness of h_gamma
  14. Proof of Part B) in Theorem \ref{['premain']}
  15. Proof of Corollary \ref{['inter']} and \ref{['main']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let Our hypotheses in A) are not scaling-invariant due to hip2main, as the set $B((N+3) \theta_a(s)^{1/2}) \setminus B(\frac{N_0}{2} \theta_{a}(s))$ is not parabolic. Thus, to address the general case, we introduce the parameters $a, b, c$, where $a$ determines the aperture of the paraboloid, $b$ sp

Figures (3)

  • Figure 1: Parabolic scales
  • Figure 2: Regions in dynamic decomposition with $a=1$ and $N=6$
  • Figure 3: Non scaling invariant region in Theorem \ref{['premain']} taking $a=1$ and $N=6$

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Corollary 2