On the Rayleigh-Taylor instability and regularity for the Vlasov system in a convex domain

We study two different subjects. The first part investigates the dynamical Rayleigh-Taylor instability for the density-dependent incompressible Euler equations in two space dimensions. It is shown that the classical Rayleigh-Taylor instability is valid in a nonlinear dynamical sense for smooth density profiles. The second part concerns the initial-boundary value problem in a convex domain for the Vlasov-Poisson system. Boundary effects play an important role in such physical problems that are modeled by the Vlasov-Poisson system. We establish the global existence of classical solutions with regular initial boundary data under the absorbing boundary condition. We also prove that regular symmetric initial data lead to unique classical solutions for all time in the case of the specularly reflecting boundary condition.