### On Instability and Stability of Gravity Driven Navier-Stokes-Korteweg Model in Two Dimensions-江飞教授（福州大学）

Title: On Instability and Stability of Gravity Driven Navier-Stokes-Korteweg Model in Two Dimensions

Speaker: 江飞 教授（福州大学）

Abstract

Bresch-Desjardins-Gisclon-Sart have formally derived that the capillarity can slow the growth rate of Rayleigh-Taylor (RT) instability in the capillary fluids based on the linearized two-dimensional (2D) Navier-Stokes-Korteweg equations in 2008. Motivated by their linear theory, we further investigate the nonlinear Rayleigh-Taylor instability problem for the 2D incompressible case in a horizontal slab domain with Navier boundary condition, and rigorously verify that the RT instability can be inhibited by capillarity under our 2D setting. More precisely, if the RT density profile $\bar{\rho}$ satisfies an additional stabilizing condition, then there is a threshold of capillarity coefficient $\kappa_C$, such that if the capillary coefficient $\kappa$ is bigger than $\kappa_C$, then the small perturbation solution around the RT equilibrium state is algebraically stable in time. In particular, if the RT density profile is linear, then the critical number can be given by the formula $\kappa_C= g h^2 /\bar{\rho}' \pi^2$, where $g$ is the gravity constant and $h$ the height of the slab domain. In addition, we also provide a nonlinear instability result for $\kappa\in[0, \kappa_C)$. The instability result presents that the capillarity can not inhibit the RT instability, if it's strength is too small. This is a joint work with Fucai Li and Zhipeng Zhang