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On rough Calderón solutions to the Navier-Stokes equations and applications to the singular set

Henry Popkin

Abstract

In 1934, Leray proved the existence of global-in-time weak solutions to the Navier-Stokes equations for any divergence-free initial data in $L^2$. In the 1980s, Giga and Kato independently showed that there exist global-in-time mild solutions corresponding to small enough critical $L^3(\mathbb{R}^3)$ initial data. In 1990, Calderón filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in $L^p$ for $2< p<3$ by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a "Calderón-like" splitting to show the global-in-time existence of weak solutions to the Navier-Stokes equations corresponding to supercritical Besov space initial data $u_0 \in \dot{B}^{s}_{q,\infty}$ where $q>2$ and $-1+\frac{2}{q}<s<\min \left(-1+\frac{3}{q},0 \right)$, which fills a similar gap between Leray and known mild solution theory in the Besov space setting. We also use the Calderón-like splitting to investigate the structure of the singular set under a Type-I blow-up assumption in the Besov space setting, which is considerably rougher than in previous works.

On rough Calderón solutions to the Navier-Stokes equations and applications to the singular set

Abstract

In 1934, Leray proved the existence of global-in-time weak solutions to the Navier-Stokes equations for any divergence-free initial data in . In the 1980s, Giga and Kato independently showed that there exist global-in-time mild solutions corresponding to small enough critical initial data. In 1990, Calderón filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in for by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a "Calderón-like" splitting to show the global-in-time existence of weak solutions to the Navier-Stokes equations corresponding to supercritical Besov space initial data where and , which fills a similar gap between Leray and known mild solution theory in the Besov space setting. We also use the Calderón-like splitting to investigate the structure of the singular set under a Type-I blow-up assumption in the Besov space setting, which is considerably rougher than in previous works.

Paper Structure

This paper contains 23 sections, 15 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Comparison with previous literature
  4. Theorem \ref{['Global solution supercritical Besov data']} and (Q1)
  5. Theorem \ref{['Type1 singular set']} and (Q2)
  6. Preliminaries
  7. General notation
  8. Besov and Kato spaces
  9. General definitions for the Navier-Stokes equations
  10. Critical quantities for the Navier-Stokes equations
  11. Leray-Hopf weak solutions
  12. The (putative) singular set
  13. Mild solution notation
  14. The $\boldsymbol{v}$-perturbed Navier-Stokes equations
  15. Calderón-type splitting of Besov Space initial data
  16. ...and 8 more sections

Key Result

Theorem A

Let $2<q<\infty$ and $-1+\frac{2}{q}<s<\min \left(-1+\frac{3}{q},0 \right).$ Suppose that $u_0 \in \dot{B}^{s}_{{q},{\infty}}(\mathbb{R}^3)$ is divergence-free. Then there exist constants $\max(q,3)< p<\infty$ and $0<\delta<1-\frac{3}{p}$ (both depending on $q$ and $s$) and a suitableAs in Definitio corresponding to an initial data splitting such that:

Figures (3)

  • Figure 1: Splitting diagram for Proposition \ref{['genBesovsplit']}This illustrates the splitting proof from AlbrBarker.
  • Figure 2: Subcritical/critical splitting diagram, corresponding to Proposition \ref{['genBesovsplit']}/Figure \ref{['splitting diagram']}. Observe that the highlighted orange region $D$ is contained fully in the subcritical region under the red line $l_{crit}$. This means that any point in $D$ would be viable for our purposes. We choose $p=2q$ and bisect between $l_1$ and $l_2$ for simplicity.
  • Figure 3: Supercritical splitting diagram, corresponding to Proposition \ref{['genBesovsplit']}/Figure \ref{['splitting diagram']}. Notice here that the highlighted orange region $D$ is not fully contained in the subcritical region under the red line $l_{crit}$. This means that we must take care to choose a point in $D$ that corresponds to a subcritical Besov space. We can see the significance of the condition $-1+\frac{2}{q}<s<0$, which corresponds to the region below the blue line $l_{bndry}$; if we have our initial data in a Besov space corresponding to a point above $l_{bndry}$, then it is clear that $l_1$ (the line defined through $\frac{1}{2}$ (corresponding to $L^2$) and $(s,q)$ (corresponding to $\dot{B}^{s}_{{q},{\infty}}$)) does not interest with $l_{crit}$, and hence the region $D$ would not contain any subcritical points.

Theorems & Definitions (32)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 22 more