On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

Abstract This paper focuses on the Rayleigh–Taylor instability in two-dimensional system of equations nonhomogeneous incompressible viscous fluids with capillarity effects a horizontal periodic domain infinite height. First, we use modified variational method to construct (linear) unstable solutions for linearized capillary problem. Then, motivated by Grenier’s idea (Grenier Commun. Pure Appl. Math. 53(9):1067–1091, 2000), further approximate higher-order growing modes problem and derive error estimates between both nonlinear Finally, prove existence escape points based bootstrap Hwang–Guo (Hwang Guo Arch. Ration. Mech. Anal. 167(3):235–253, 2003), thus obtain result. Our result presents that can occur any coefficient $\kappa >0$ κ > 0 if critical is infinite. In particular, it improves previous Zhang’s (Zhang J. Fluid 24(3):70–23, 2022) assumption smallness coefficient.