Rayleigh-Taylor instability for incompressible viscous quantum flows

Abstract

Consider the linear and nonlinear Rayleigh-Taylor instability of the three-dimensional incompressible viscous Navier-Stokes-Quantum equations. For linearized problem, we determine the critical number κc precisely. Then, we construct a linear growth solution by a modified variational method for κ<κc. In addition, we show that κc is infinite for a special steady state ρ¯, which implies that quantum potential inhibit the instability instead of cutting it off. Based on this unstable linear solution and the priori estimates of the smooth solution to the perturbed problem, we establish the nonlinear instability of the density and the velocities in the sense of Hadamard. Compared with the related study on Navier-Stokes-Korteweg equations (Zhang (2022) [41]), we do not ask the capillarity coefficient to be small.