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Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system

David Barbato, Francesco Morandin, Marco Romito

Abstract

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009].

Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system

Abstract

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009].

Paper Structure

This paper contains 8 sections, 19 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. From the generalized Fourier Navier--Stokes to the dyadic equation
  3. The shell approximation
  4. The shell solution
  5. The shell model as a shell approximation
  6. From the dyadic equation to the recursive inequality
  7. Solving the recursion
  8. Local existence and uniqueness

Key Result

Theorem 1.1

Let $d\geq 2$ and assume e:asstao1, e:asstao3 and e:assbmr for a non--decreasing function $G:[0,\infty)\to[0,\infty)$. Then e:gnse has a global smooth solution for every smooth initial condition.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4: shell solution
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 31 more