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On the Inhibition of Rayleigh Taylor Instability by Capillarity in the Navier Stokes Korteweg Model

Fei Jiang, Yajie Zhang, Zhipeng Zhang

Abstract

Bresch--Desjardins--Gisclon--Sart had derived that the capillarity slows down the growth rate of Rayleigh--Taylor (RT) instability in an inhomogeneous incompressible fluid endowed with internal capillarity based on a linearized incompressible Navier--Stokes--Korteweg (NSK) equations in 2008. Later Li--Zhang further obtained another result that the capillarity inhibits RT instability also based on the linearized equations in (SIAM J. Math. Anal. 3287--3315, 2023), if the capillarity coefficient is bigger than some threshold. In this paper, we further rigorously prove such phenomenon of capillarity inhibiting the RT instability in the \emph{nonlinear} incompressible NSK equations in a horizontally periodic slab domain with Navier (slip) boundary conditions. The key idea in the proof is to capture the dissipative estimates of the tangential derivatives of density. Such dissipative estimates result in the decay-in-time of both the velocity and the perturbation density which is very useful to overcome the difficulties arising from the nonlinear terms.

On the Inhibition of Rayleigh Taylor Instability by Capillarity in the Navier Stokes Korteweg Model

Abstract

Bresch--Desjardins--Gisclon--Sart had derived that the capillarity slows down the growth rate of Rayleigh--Taylor (RT) instability in an inhomogeneous incompressible fluid endowed with internal capillarity based on a linearized incompressible Navier--Stokes--Korteweg (NSK) equations in 2008. Later Li--Zhang further obtained another result that the capillarity inhibits RT instability also based on the linearized equations in (SIAM J. Math. Anal. 3287--3315, 2023), if the capillarity coefficient is bigger than some threshold. In this paper, we further rigorously prove such phenomenon of capillarity inhibiting the RT instability in the \emph{nonlinear} incompressible NSK equations in a horizontally periodic slab domain with Navier (slip) boundary conditions. The key idea in the proof is to capture the dissipative estimates of the tangential derivatives of density. Such dissipative estimates result in the decay-in-time of both the velocity and the perturbation density which is very useful to overcome the difficulties arising from the nonlinear terms.
Paper Structure (10 sections, 24 theorems, 198 equations)

This paper contains 10 sections, 24 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Mathematical formulation for the capillary RT problem
  3. Notations
  4. A stability result
  5. A priori estimates
  6. Preliminaries
  7. A total energy inequality
  8. Tangential energy inequality with decay-in-time
  9. Proof of Theorem \ref{['thm2']}
  10. Analysis tools

Key Result

Theorem 1.1

Let $\mu$ and $\kappa$ be positive constants. If $\kappa$ and $\bar{\rho}\in {C^7}[0,h]$ satisfy 0102, the sharp stability condition 2020102241504, the stabilizing condition 2022205071434, and the additional boundary condition of the density profile there is a sufficiently small constant $\delta\in (0,1)$, such that for any $( \varrho^0,v^0)\in {H}^{4}_{\bar{\rho}} \times {^0_\sigma {{H}}^3_{\mat

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 2.1
  • Lemma 2.8
  • ...and 14 more