On 2D Rayleigh-Taylor instabilities

1 Introduction Consider in the plane the flow of two ideal incompressible fluids of constant densities ρ ± > 0 in the gravity field, with ρ + = ρ −. Velocity field satisfies the Euler equation ∂ρu ∂t + u · ∇ρu = −∇p + ρg (1) div u = 0 (2) ρ t + div (ρu) = 0 (3) with initial data u(x, 0) = u 0 (x). (4) We suppose, that u 0 (x) satisfies the continuity equation (2), and the assumption that vorticity ω 0 (x) = rot u 0 is concentrated on some curve Σ 0 separating the two fluids, i.e. we suppose Σ 0 splits the plane into two domains Ω 0 + and Ω 0 − , in which the fluids have constant densities ρ + and ρ − respectively. So we have ω 0 = Ωδ Σ 0 (5) with Ω = [u || ] being the jump of the tangent to Σ 0 component of velocity. Applying the rot operator to (1) we find that vorticity ω = ∂ x u y − ∂ y u x is formally constant along the flow in each subdomain where ρ is constant : ∂ t ω + u · ∇ω = 0 (6) Thus for t > 0 one expects the vorticity to remain concentrated on some curve Σ t , with a time dependant curve Σ t separating the two fluids. This problem is known as the Rayleigh-Taylor instability. It has been shown in [SS85] that in the case of a periodic interface close to a flat line and with analytic data Ω, Σ 0 , this problem is locally in time well posed. On the other hand, in [Leb02] and [Wu], it has been proved that for the Kelvin-Helmholtz problem, i.e in the case ρ + = ρ − , g = 0 the evolution problem of the vortex sheet is strongly ill-posed in the sense that if Ω does not vanish, analyticity of the data is a necessary condition to get a local in time solution with a C 1+α interface Σ with α > 0. The main goal of this paper is to extend the above results on Kelvin-Helmholtz instability to the more involved Rayleigh-Taylor case. In order to fix geometry, we suppose that the interface Σ t is a closed simple curve in the plane (the case of periodic …