Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear Stability of the Rayleigh-Taylor Problem in Quantum Navier-Stokes Equations

Fei Jiang, Yajie Zhang, Zhipeng Zhang, Youyi Zhao

TL;DR

The paper tackles nonlinear Rayleigh–Taylor instability for the quantum Navier–Stokes system in a slab, showing that quantum stabilization via the Bohm potential yields nonlinear stability when the scaled Planck constant $\varepsilon$ exceeds a finite threshold $\varepsilon_c$, under a stabilizing density profile. The authors develop a novel multi-layer energy method with anisotropic spatial norms, establishing a sequence of a priori estimates that control high-order, tangential, and negative-derivative quantities and yield algebraic decay. A bootstrap argument closes the estimates, proving global existence and decay of small perturbations around the RT equilibrium, with the density remaining strictly positive. The results provide a rigorous nonlinear counterpart to linear quantum stabilization and offer a robust energy-analytic framework applicable to quantum fluids with RT-type instabilities and Navier boundary conditions.

Abstract

It is well-known that the Rayleigh--Taylor (abbr. RT) instability can be completely inhibited by the quantum effect stabilization in proper circumstances leading to a cutoff wavelength in the \emph{linear} motion equations. Motivated by the linear theory, we further investigate the {stability} for the \emph{nonlinear} RT problem of quantum Navier--Stokes equations in a slab with Navier boundary condition, and rigorously prove the inhibition of RT instability by the quantum effect under a proper setting. More precisely, if the RT density profile $\barρ$ satisfies an additional stabilizing condition, then there is a threshold ${\varepsilon_{c}}$ of the scaled Planck constant, such that if the scaled Planck constant is bigger than ${\varepsilon_{c}}$, the small perturbation solutions around an RT equilibrium state are algebraically stable in time. The mathematical proof is realized by a complicated multi-layer energy method with anisotropic norms of spacial derivatives.

Nonlinear Stability of the Rayleigh-Taylor Problem in Quantum Navier-Stokes Equations

TL;DR

The paper tackles nonlinear Rayleigh–Taylor instability for the quantum Navier–Stokes system in a slab, showing that quantum stabilization via the Bohm potential yields nonlinear stability when the scaled Planck constant exceeds a finite threshold , under a stabilizing density profile. The authors develop a novel multi-layer energy method with anisotropic spatial norms, establishing a sequence of a priori estimates that control high-order, tangential, and negative-derivative quantities and yield algebraic decay. A bootstrap argument closes the estimates, proving global existence and decay of small perturbations around the RT equilibrium, with the density remaining strictly positive. The results provide a rigorous nonlinear counterpart to linear quantum stabilization and offer a robust energy-analytic framework applicable to quantum fluids with RT-type instabilities and Navier boundary conditions.

Abstract

It is well-known that the Rayleigh--Taylor (abbr. RT) instability can be completely inhibited by the quantum effect stabilization in proper circumstances leading to a cutoff wavelength in the \emph{linear} motion equations. Motivated by the linear theory, we further investigate the {stability} for the \emph{nonlinear} RT problem of quantum Navier--Stokes equations in a slab with Navier boundary condition, and rigorously prove the inhibition of RT instability by the quantum effect under a proper setting. More precisely, if the RT density profile satisfies an additional stabilizing condition, then there is a threshold of the scaled Planck constant, such that if the scaled Planck constant is bigger than , the small perturbation solutions around an RT equilibrium state are algebraically stable in time. The mathematical proof is realized by a complicated multi-layer energy method with anisotropic norms of spacial derivatives.

Paper Structure

This paper contains 15 sections, 24 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. A quantum RT problem
  3. Notations
  4. A stability result
  5. Sketch the proof for the stability result
  6. A priori estimates
  7. Preliminaries
  8. A basic energy estimate and a stabilizing estimate
  9. Estimates of high-order spacial derivatives
  10. Tangential estimates
  11. Dissipative estimates of $\varrho$
  12. Estimates of temporal derivatives
  13. Energy inequalities
  14. Proof of Theorem \ref{['thm2']}
  15. Appendix: analysis tools

Key Result

Theorem 1.1

Let $\mu$, $\varepsilon$ be positive constants, and $\bar{\rho}\in {C^8}[0,h]$ satisfy the stabilizing condition 2022205071434 and the boundary condition Under the sharp stability condition 2saf01504, there is a sufficiently small constant $\delta\in (0,1)$, such that for any $( \varrho^0,v^0)\in { {H}}_{-s,3}^{\bar{\rho},6}\times {H}_{-s,2}^{\diamond,5}$ satisfying necessary compatibility con

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
  • Lemma 2.8
  • Lemma 2.9
  • ...and 14 more