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A note on certain scenarios of Type II blowups of suitable weak solutions to the Navier-Stokes equations

Gregory Seregin

Abstract

In the note, various scenarios of potential Type II blowups of suitable weak solutions to the Navier-Stokes equations are studied. It is shown, that under some assumptions, such type of blowups cannot happen. In this case, corresponding statements may be interpreted as regularity results. Their justification is based on a technique making use of a certain Euler scaling and Liouville type theorems for ancient solutions to the Euler system.

A note on certain scenarios of Type II blowups of suitable weak solutions to the Navier-Stokes equations

Abstract

In the note, various scenarios of potential Type II blowups of suitable weak solutions to the Navier-Stokes equations are studied. It is shown, that under some assumptions, such type of blowups cannot happen. In this case, corresponding statements may be interpreted as regularity results. Their justification is based on a technique making use of a certain Euler scaling and Liouville type theorems for ancient solutions to the Euler system.

Paper Structure

This paper contains 5 sections, 5 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Impossible Cases of Potential Type II Blowups in Scenario \ref{['TypeII']} and \ref{['M1']}
  3. One More Scenario of Type II Blowup
  4. Some Liouville Type Theorem for the Euler system
  5. Ladyzhenskaya-Prodi-Serrin Type Condition

Key Result

Theorem 2.1

Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q$. It is supposed that the pair obeys condition M1. Assume further that, for some numbers $s$ and $l$, satisfying restriction (restriction on s and l), the additional inequalities hold with some parameter $0\leq\eta \leq 1$. Suppose that where $\alpha=2-m$. Then

Theorems & Definitions (10)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.2
  • proof
  • Theorem 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 5.1
  • Remark 5.2
  • proof