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

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 kappa > 0 if the critical capillary number is infinite. (c) 2022 Elsevier Inc. All rights reserved.