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

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 $\bar\rho$ 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.