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A nonlinear instability result to the Navier-Stokes equations with Navier slip boundary conditions

Tien-Tai Nguyen

TL;DR

This work analyzes the nonlinear instability of the trivial steady state for the incompressible Navier–Stokes equations in a slab with Navier–slip boundary conditions. It proves linear instability by constructing an infinite sequence of unstable normal modes through an operator framework, valid in the subcritical viscosity regime $\mu<\mu_c(k,\Xi)$ and globally for $\mu<\mu_c(\Xi)$. Building on these modes, it establishes nonlinear instability in the Desjardins Grenier sense by forming a linear combination of modes and controlling the nonlinear remainder to yield two solutions that separate at an escaping time $T^\delta$. The results illuminate the role of slip coefficients $\xi_\pm$ in destabilizing shear flows and offer a distinct approach from prior work such as DLX18, advancing understanding of boundary-slip effects on viscous instability.

Abstract

In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).

A nonlinear instability result to the Navier-Stokes equations with Navier slip boundary conditions

TL;DR

This work analyzes the nonlinear instability of the trivial steady state for the incompressible Navier–Stokes equations in a slab with Navier–slip boundary conditions. It proves linear instability by constructing an infinite sequence of unstable normal modes through an operator framework, valid in the subcritical viscosity regime and globally for . Building on these modes, it establishes nonlinear instability in the Desjardins Grenier sense by forming a linear combination of modes and controlling the nonlinear remainder to yield two solutions that separate at an escaping time . The results illuminate the role of slip coefficients in destabilizing shear flows and offer a distinct approach from prior work such as DLX18, advancing understanding of boundary-slip effects on viscous instability.

Abstract

In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).
Paper Structure (11 sections, 7 theorems, 51 equations)

This paper contains 11 sections, 7 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. The goal of this paper
  4. Main results
  5. The linear instability
  6. Auxiliary operators
  7. Normal mode solutions
  8. The nonlinear instability
  9. Linear combination of normal modes
  10. The difference function
  11. Proof of Theorem \ref{['ThmNonlinear']}

Key Result

Proposition 2.1

Let $\tilde{H}^s((-1,1))= \{\phi \in H^s((-1,1)), \phi(\pm 1)=0\}$ with $s\geq1$. The following results hold.

Theorems & Definitions (11)

  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of Theorem \ref{['ThmSpectral']}
  • Proposition 3.3
  • proof
  • ...and 1 more