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

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.

报告人简介Introduction to the Speaker：
    江飞教授，硕博连读于厦门大学数学科学学院，曾在北京应用物理与计算数学研究所做两年博士后，2012年9月入职福州大学，并于2017年7月应聘为教授及博士生导师。目前主要研究流体动力学中各类偏微分方程组的适定性问题及解的性态。承担过国家青年、面上及优青项目各一项，福建省面上、高校杰青、自然科学基金杰青及重点项目各一项，已在《Adv. Math.》，《Arch. Rational Mech. Anal.》，《J. Math. Pures Appl.》，《Comm. Partial Differential Equations》，《Calc. Var. Partial Differential Equations》等杂志上发表数学论文40余篇。