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Restoring the Navier--Stokes dynamics by determining functionals depending on pressure

A. Ilyin, V. Kalantarov, A. Kostianko, S. Zelik

Abstract

For 2D Navier--Stokes equations in a bounded smooth domain, we construct a system of determining functionals which consists of $N$ linear continuous functionals which depend on pressure $p$ only and of one extra functional which is given by the value of vorticity at a fixed point $x_0\in\partialΩ$.

Restoring the Navier--Stokes dynamics by determining functionals depending on pressure

Abstract

For 2D Navier--Stokes equations in a bounded smooth domain, we construct a system of determining functionals which consists of linear continuous functionals which depend on pressure only and of one extra functional which is given by the value of vorticity at a fixed point .
Paper Structure (5 sections, 11 theorems, 63 equations)

This paper contains 5 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries I. 2D Navier--Stokes equations: well-posedness, attractors and dimensions
  3. Preliminaries II. Determining and separating functionals
  4. Main result
  5. Concluding remarks and open problems

Key Result

Theorem 2.1

Let $g\in L^2(\Omega)$ and $u_0\in H$. Then, for any $T>0$, problem 1.NS possesses a unique weak solution $u(t)$ and therefore the solution semigroup is well-defined in the phase space $H$. Moreover, this semigroup is dissipative, i.e., where $\alpha>0$ and the monotone function $Q$ are independent of $t$, $u_0$ and $g$. Furthermore, this semigroup possesses the parabolic smoothing property, i.e

Theorems & Definitions (23)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 13 more