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On a 3D Stokes eigenvalue problem under Navier slip-with-friction boundary conditions and applications to Navier-Stokes equations

Luigi C. Berselli, Alessio Falocchi, Rossano Sannipoli

TL;DR

This work provides a precise spectral analysis of the Stokes operator under Navier slip-with-friction boundary conditions in a doubly periodic 3D channel and leverages the explicit eigenstructure to construct infinite-dimensional families of global smooth solutions for the 3D Navier–Stokes equations. The main contributions are the explicit characterization of eigenvalues and eigenfunctions via transcendental equations depending on the friction parameter $\beta$, the analysis of the $\beta=0$ and $\beta\to\infty$ limits, and the demonstration of global regularity for large classes of initial data built from eigenmodes. These results yield rigorous benchmarks for DNS/LES in channel-like geometries and shed light on boundary-induced effects on stability and turbulence onset in fluid flows.

Abstract

In this paper we consider, by means of a precise spectral analysis, the 3D Navier-Stokes equations endowed with Navier slip-with-friction boundary conditions. We study the problem in a very simple geometric situation as the region between two parallel planes, with periodicity along the two planes. This setting, which is often used in the theory of boundary layers, requires some special treatment for what concerns the functional setting and allows us to characterize in a rather explicit manner eigenvalues and eigenfunctions of the associated Stokes problem. These, will be then used in order to identify infinite dimensional classes of data leading to global strong solutions for the corresponding evolution Navier-Stokes equations.

On a 3D Stokes eigenvalue problem under Navier slip-with-friction boundary conditions and applications to Navier-Stokes equations

TL;DR

This work provides a precise spectral analysis of the Stokes operator under Navier slip-with-friction boundary conditions in a doubly periodic 3D channel and leverages the explicit eigenstructure to construct infinite-dimensional families of global smooth solutions for the 3D Navier–Stokes equations. The main contributions are the explicit characterization of eigenvalues and eigenfunctions via transcendental equations depending on the friction parameter , the analysis of the and limits, and the demonstration of global regularity for large classes of initial data built from eigenmodes. These results yield rigorous benchmarks for DNS/LES in channel-like geometries and shed light on boundary-induced effects on stability and turbulence onset in fluid flows.

Abstract

In this paper we consider, by means of a precise spectral analysis, the 3D Navier-Stokes equations endowed with Navier slip-with-friction boundary conditions. We study the problem in a very simple geometric situation as the region between two parallel planes, with periodicity along the two planes. This setting, which is often used in the theory of boundary layers, requires some special treatment for what concerns the functional setting and allows us to characterize in a rather explicit manner eigenvalues and eigenfunctions of the associated Stokes problem. These, will be then used in order to identify infinite dimensional classes of data leading to global strong solutions for the corresponding evolution Navier-Stokes equations.
Paper Structure (22 sections, 21 theorems, 235 equations, 1 figure, 2 tables)

This paper contains 22 sections, 21 theorems, 235 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional setting and notations
  4. Preliminary results
  5. Main results
  6. Spectral analysis
  7. Global smooth solutions to the evolution problem
  8. Proofs of the preliminary results
  9. Proof of Proposition \ref{['lem:gradsym']}
  10. Proof of Proposition \ref{['lemma:regularity']}
  11. Proof of Proposition \ref{['lemma0']}
  12. Proof of Proposition \ref{['existence']}
  13. Proofs of the main results
  14. Proof of Theorem \ref{['eigcube']}
  15. Proof of Corollary \ref{['corollary']}
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $\beta \ge 0$ then $\|\mathbb{ D} \cdot\|_{L^2(\Omega)}$ and $\|\nabla\cdot\|_{L^2(\Omega)}$ are equivalent semi-norms (norms if $\beta>0$) in $V_\mathcal{P}$, i.e. it holds

Figures (1)

  • Figure 1: On the left the first eigenvalues $\lambda_k(\beta)$ when $\nabla q\equiv 0$ for 4 different $\beta$, on the right the first eigenvalues $\Lambda_k(\beta)$ when $\nabla q\not\equiv 0$ for 3 $\beta$; in the order from below stars $\beta=0$, triangles $\beta=1$, circles $\beta=10$ and squares $\beta\rightarrow\infty$.

Theorems & Definitions (31)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.3: Navier frictionless
  • Corollary 3.4: Dirichlet
  • ...and 21 more