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Strong solutions and attractor dimension for 2D NSE with dynamic boundary conditions

Dalibor Pražák, Michael Zelina

Abstract

We consider incompressible Navier-Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.

Strong solutions and attractor dimension for 2D NSE with dynamic boundary conditions

Abstract

We consider incompressible Navier-Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.
Paper Structure (19 sections, 26 theorems, 238 equations)

This paper contains 19 sections, 26 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Function spaces
  5. Weak formulation
  6. Dynamical systems
  7. Stokes system
  8. Eigenvalue problem - ON basis
  9. Stokes problem - stationary
  10. Stokes problem -- evolutionary
  11. Regularity for non-linear systems
  12. Existence of the solution
  13. Regularity for NS system
  14. Regularity for systems with quadratic growth
  15. Dimension of the attractor
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let us consider the system eq:Equation--eq:DerivativeBoundary with $\Omega \in \mathcal{C}^{1, 1}$ and the initial condition $\bm u_0 \in H$. Concerning the Cauchy stress we further suppose that and hold for all symmetrical $2\times 2$ matrices $\bm D, \bm E$. Concerning the boundary non-linearity we require that Let $2 < p < +\infty$ be given, we denote and suppose Then there is $q > 2$ such

Theorems & Definitions (61)

  • Theorem 1.1: Strong solutions
  • Remark
  • Remark
  • Theorem 1.2: Dimension estimate
  • Lemma 1.3
  • proof
  • Remark
  • Definition
  • Theorem 2.1: Basis of $V$
  • proof
  • ...and 51 more