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

2025/10/09 by Fei Jiang, Jiang, Fei, Yajie Zhang +5
Mathematics · #35Q35 #76D03 #76E99 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2510.07695

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

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 linear motion equations. Motivated by the linear theory, we further investigate the stability for the 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 ρ satisfies an additional stabilizing condition, then there is a threshold ε_c of the scaled Planck constant, such that if the scaled Planck constant is bigger than ε_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.

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