Rayleigh–Taylor instability in stratified compressible fluids with/without the interfacial surface tension

Source abstract

Abstract Guo–Tice formally established in 2011 that the linear Rayleigh–Taylor instability inevitably occurs within stratified compressible viscous fluids in a slab domain , irrespective of the presence of interfacial surface tension, where the unstable solutions are nonperiodic with respect to both horizontal spatial variables and , by applying a so‐called “normal mode” method and a modified variational method to the linearized (motion) equations (Guo and Tice (SIAM J. Math. Anal. 42 (2011), 1688–1720)]. It remains an open problem, however, whether Guo–Tice's conclusion can be rigorously verified by the (original) nonlinear equations. The mathematical difficulty arises due to the failure of constructing a growing mode solution, which is nonperiodic with respect to both horizontal spatial variables, to the linearized equations defined on a slab domain. In the present work, we circumvent the difficulty related to growing mode solutions by developing an alternative approximate approach. In essence, our approach hinges on constructing the horizontally periodic growing mode solution of the linearized equations to approximate the nonlinear Rayleigh–Taylor unstable solutions, which do not exhibit horizontal periodicity. Thanks to this new approximate approach, we can apply Guo–Hallstrom–Spirn's bootstrap (instability) method by Guo et al. [Commun. Math. Phys. 270 (2007), 635–689] to the nonlinear equations in Lagrangian coordinates, and thus prove Guo–Tice's conclusion.