國立清華大學 數學系

Abstract: In this talk, we introduce mathematical progress on nonlinear Rayleigh-Taylor instability in fluids. In particular, we will report our last results on instability and stability of some steady-states of a three-dimensional viscous flow driven by gravity in a bounded domain. More precisely, (1) If the steady density is heavier with increasing height (i.e., the Rayleigh-Taylor steady-state) for the both cases of nonhomogeneous incompressible fluids and compressible fluids without heat conductivity, we show that the steady-state is linear unstable (i.e., the linear solution grows in time) by exploiting the modified variational method. Then, by developing some new ideas and using a careful bootstrap argument, we further show that the steady-state is nonlinear unstable in the sense of Hadamard. (2) If the steady density is lighter with increasing height for nonhomogeneous incompressible fluids, we show, with the help of a restricted condition imposed on steady density, that the steady-state is linearly globally stable and nonlinearly locally stable in the sense of Hadamard. Tea Time: 3:30PM, R707