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arXiv · 2510.07695

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

Fei Jiang·Yajie Zhang·Zhipeng Zhang·Youyi Zhao
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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.

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Fei Jiang, Yajie Zhang, Zhipeng Zhang, Youyi Zhao. 2025-10-09. Nonlinear Stability of the Rayleigh-Taylor Problem in Quantum Navier-Stokes Equations. https://arxiv.org/abs/2510.07695

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