Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forced Rapidly Dissipative Navier--Stokes Flows

Lorenzo Brandolese, Takahiro Okabe

Abstract

We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier--Stokes equations in $\mathbb{R}^n$ . The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.

Forced Rapidly Dissipative Navier--Stokes Flows

Abstract

We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier--Stokes equations in . The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.

Paper Structure

This paper contains 9 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Overview of the proof of Theorem \ref{['th:theo1']}.
  4. Constructing solutions in $L^2(\mathbb{R}^n\times \mathbb{R}^+)$
  5. Revisiting Fujigaki and Miyakawa asymptotic profiles
  6. Rapidly dissipative heat flows
  7. Constructing $f$ and a solution $v$ with a prescribed energy matrix
  8. Conclusion of the proof of Theorem \ref{['th:theo1']}
  9. Acknowledgements

Key Result

Theorem 2.1

Let $2\le n<r<\infty$. Let Let $\chi\in L^\infty_c(\mathbb{R}^n\times\mathbb{R}^+)$, such that $\int_0^\infty\!\!\int\chi=1$. There exist $\eta_0>0$ (only dependent on $n$ and $r$), and a constant real matrix $(\sigma_{k\ell})$ (dependent on $n$, $r$ and also on $a$, $\chi$), with such that if and if then there exists a global solution $u\in X_r\cap L^2(\mathbb{R}^n\times\mathbb{R}^+)$ to (NS)

Theorems & Definitions (19)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • proof
  • Proposition 3.1
  • ...and 9 more