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Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier-Stokes equations on the torus

David M. Ambrose, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes

Abstract

We consider the three-dimensional Navier-Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier-Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity.

Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier-Stokes equations on the torus

Abstract

We consider the three-dimensional Navier-Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier-Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity.
Paper Structure (8 sections, 4 theorems, 63 equations)

This paper contains 8 sections, 4 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of solutions
  4. Linear estimates
  5. Bilinear estimates
  6. Analyticity of solutions
  7. Linear estimates for analyticity
  8. Bilinear estimates for analyticity

Key Result

Lemma 1

There exists $c>0$ such that for all $k\in\mathbb{Z}^{3}_{*},$

Theorems & Definitions (12)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • ...and 2 more