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On the absence of anomalous dissipation for the Navier-Stokes equations with Navier boundary conditions: a sufficient condition

Claude Bardos, Daniel W. Boutros, Edriss S. Titi

Abstract

We consider the three-dimensional incompressible Navier-Stokes equations in a bounded domain with Navier boundary conditions. We provide a sufficient condition for the absence of anomalous energy dissipation without making assumptions on the behaviour of the corresponding pressure near the boundary or the existence of a strong solution to the incompressible Euler equations with the same initial data. We establish our result by using our recent regularity results for the pressure corresponding to weak solutions of the incompressible Euler equations [Arch. Ration. Mech. Anal., 249 (2025), 28].

On the absence of anomalous dissipation for the Navier-Stokes equations with Navier boundary conditions: a sufficient condition

Abstract

We consider the three-dimensional incompressible Navier-Stokes equations in a bounded domain with Navier boundary conditions. We provide a sufficient condition for the absence of anomalous energy dissipation without making assumptions on the behaviour of the corresponding pressure near the boundary or the existence of a strong solution to the incompressible Euler equations with the same initial data. We establish our result by using our recent regularity results for the pressure corresponding to weak solutions of the incompressible Euler equations [Arch. Ration. Mech. Anal., 249 (2025), 28].

Paper Structure

This paper contains 5 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminary results and weak formulation of the Euler and Navier-Stokes equations
  3. Main result and its proof
  4. Concluding remarks
  5. Global existence of Leray-Hopf weak solutions for generalised Navier boundary conditions

Key Result

Proposition 5

Let $\alpha \in (0,1)$, and let $u \in C([0,T];C^{0,\alpha} (\Omega))$ be weakly divergence-free and satisfy $(u \cdot n) \lvert_{\partial \Omega} = 0$ as well as $u \lvert_{t = 0} = u_0$ (where $u_0 \in C^{0,\alpha} (\Omega)$ is divergence-free). Then the weak formulation eulerweakformulation is eq for all divergence-free $\psi \in \mathcal{D} (\Omega \times [0,T); \mathbb{R}^3)$. To be more prec

Theorems & Definitions (12)

  • Definition 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Remark 7
  • Remark 8
  • ...and 2 more