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Well-posed Self-Similarity in Incompressible Standard Flows

J. Polihronov

TL;DR

The paper addresses global regularity of the 3D incompressible Navier–Stokes equations by exploiting self-similarity and Bouton-Lie invariants to derive a universal self-similar form. It shows that, under standard flow scaling, certain non-standard isobaric-weight self-similar solutions can embed space-periodic or Schwartz-class initial data and remain smooth for all time with uniformly bounded energy and vorticity, verified via Beale–Kato–Majda criteria. The main contribution is a global-regularity result for these data subclasses through embedding into a self-similar family and the associated a priori bounds, positioning symmetry-based methods as a constructive complement to classical PDE theory. However, the results do not cover arbitrary non-self-similar initial data, highlighting an important area for further investigation. Overall, the work clarifies how isobaric-weight constraints and scaling invariants can control potential blowup, offering a pathway to global regularity for a broad, physically relevant subset of NSE initial conditions.

Abstract

This article reviews the properties of the self-similar solutions of the Navier-Stokes equation for incompressible fluids. Since any smooth solution can be embedded into a self-similar solution at the identity scale, it follows that under standard flow conditions, the initial solution will remain smooth for all time as long as the self-similar solution is selected to have certain isobaric weight.

Well-posed Self-Similarity in Incompressible Standard Flows

TL;DR

The paper addresses global regularity of the 3D incompressible Navier–Stokes equations by exploiting self-similarity and Bouton-Lie invariants to derive a universal self-similar form. It shows that, under standard flow scaling, certain non-standard isobaric-weight self-similar solutions can embed space-periodic or Schwartz-class initial data and remain smooth for all time with uniformly bounded energy and vorticity, verified via Beale–Kato–Majda criteria. The main contribution is a global-regularity result for these data subclasses through embedding into a self-similar family and the associated a priori bounds, positioning symmetry-based methods as a constructive complement to classical PDE theory. However, the results do not cover arbitrary non-self-similar initial data, highlighting an important area for further investigation. Overall, the work clarifies how isobaric-weight constraints and scaling invariants can control potential blowup, offering a pathway to global regularity for a broad, physically relevant subset of NSE initial conditions.

Abstract

This article reviews the properties of the self-similar solutions of the Navier-Stokes equation for incompressible fluids. Since any smooth solution can be embedded into a self-similar solution at the identity scale, it follows that under standard flow conditions, the initial solution will remain smooth for all time as long as the self-similar solution is selected to have certain isobaric weight.
Paper Structure (28 sections, 7 theorems, 104 equations)

This paper contains 28 sections, 7 theorems, 104 equations.

Table of Contents

  1. Self-similar solutions of the NSE
  2. Standard Flows
  3. Global Regularity via Self-Similarity
  4. Comparison with Classical and Modern Results
  5. Detailed Proof of Theorem \ref{['T2']}
  6. Detailed Proof of Theorem \ref{['T3']} and Corollary \ref{['C3']}
  7. Examples of Solutions in the $U F\left( \frac{x}{L}, \frac{t}{T} \right)$ Form
  8. Example: Periodic, Smooth Initial Data
  9. Example: Schwartz‐Class Initial Data
  10. 1. Self‐similar profile and embedding at $k=1$.
  11. 2. Time‐derivative.
  12. 3. Convective term.
  13. 4. Viscous term.
  14. 5. Incompressibility.
  15. 6. Momentum equation and pressure.
  16. ...and 13 more sections

Key Result

Lemma 1.1

When the self-similar solution (gen-selfsimilarsol1) with isobaric weight $\beta_x-\beta_t$ rescales in standard flow conditions where $\alpha_x=1, \alpha_t=2$, the solution is not subject to scaling induced blowup of the velocity, the energy or the vorticity as long as

Theorems & Definitions (13)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • Theorem 3.1: Global Regularity via Self-Similarity
  • proof
  • Theorem A.1
  • proof
  • Theorem B.1
  • ...and 3 more