On the dynamic Rayleigh–Taylor instability in the Euler–Korteweg model

Abstract

This paper focuses on the Rayleigh–Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible Euler–Korteweg equations in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier's idea in [21], we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in [28], and thus obtain the nonlinear Rayleigh–Taylor instability result, which presents that the Rayleigh–Taylor instability can occur in the capillary fluids for any capillary coefficient κ>0 if the critical capillary number is infinite.