On the inhibition of Rayleigh–Taylor instability by surface tension in stratified incompressible viscous fluids under Lagrangian coordinates

The inhibition phenomenon of Rayleigh–Taylor instability by surface tension in stratified incompressible viscous fluids driven by gravity has been established in [Y.J. Wang, I. Tice and C. Kim, Arch. Ration. Mech. Anal., 212 (2014), pp. 1–92] via a special flattening coordinate transformation. However, it remains an open problem whether this inhibition phenomenon can be rigorously verified in Lagrangian coordinates due to the delicate nonlinear part of the surface tension term. In this paper, we provide a new mathematical approach, together with some key observations, to prove that the Rayleigh–Taylor problem in Lagrangian coordinates admits a unique global-in-time solution under the sharp stability condition $ \vartheta>\vartheta_{\mathrm{C}} $, where $ \vartheta $ and $ \vartheta_{\mathrm{C}} $ are the surface tension coefficient and the threshold of the surface tension coefficient, respectively. Furthermore, the solution decays exponentially in time to the equilibrium. Our result provides a rigorous proof of the inhibition phenomenon of Rayleigh–Taylor instability by surface tension under Lagrangian coordinates.