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Highly singular (frequentially sparse) steady solutions for the 2D Navier-Stokes equations on the torus

Pierre Gilles Lemarié-Rieusset

Abstract

We construct non-trivial steady solutions in $H^{-1}$ for the 2D Navier-Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to redefine the notion of solutions.

Highly singular (frequentially sparse) steady solutions for the 2D Navier-Stokes equations on the torus

Abstract

We construct non-trivial steady solutions in for the 2D Navier-Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to redefine the notion of solutions.
Paper Structure (4 sections, 3 theorems, 71 equations)

This paper contains 4 sections, 3 theorems, 71 equations.

Table of Contents

  1. Steady solutions for the Navier-Stokes problem on $\mathbb{T}^d$: known results.
  2. Admissible vector fields.
  3. 2D steady solutions.
  4. A remark on the Koch--Tataru theorem.

Key Result

Theorem 1

There exists non-trivial solutions to the equations where $\vec{u}$ is an admissible vector field (with mean value $0$) with $\vec{u}\in H^{-1}(\mathbb{T}^2)\cap BMO^{-1}$.

Theorems & Definitions (6)

  • Definition 1: Admissible vector fields
  • Theorem 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof