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

Abstract

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

Keywords

• Incompressible viscous fluids with capillarity effects
• Rayleigh–Taylor instability
• Incompressible Navier–Stokes–Korteweg equations