The linear and nonlinear Rayleigh–Taylor instability for the quasi-isobaric profile

We study the 2D system of incompressible gravity driven Euler equations in the neighborhood of a particular smooth density profile ρ 0 ( x ) such that ρ 0 ( x ) = ρ a ξ ( x L 0 ) , where ξ is a nonconstant solution of ξ ̇ = ξ ν + 1 ( 1 − ξ ) , L 0 > 0 is the width of the ablation region, ν > 1 is the thermal conductivity exponent, and ρ a > 0 is the maximum density of the fluid. The linearization of the equations around the stationary solution ( ρ 0 , 0 → , p 0 ) , ∇ p 0 = ρ 0 g → leads to the study of the Rayleigh equation for the perturbation of the velocity at the wavenumber k : − d d x ( ρ 0 ( x ) d u ¯ d x ) + k 2 ( ρ 0 ( x ) − g γ 2 ρ 0 ′ ( x ) ) u ¯ = 0 . We denote by the terms ‘eigenmode and growth rate’ an L 2 ( R ) solution of the Rayleigh equation associated with a value of γ . The purpose of this paper is twofold: • the following expansion in k L 0 , for small k L 0 , of the unique reduced linear growth rate γ g k ∈ [ 1 4 , 1 ] g k γ 2 = 1 + 2 Γ ( 1 + 1 ν ) ( 2 k L 0 ν ) 1 ν + a 2 ( k L 0 ) 2 ν + O ( k L 0 ) where a 2 is explicitly known, provided ν > 2 , the nonlinear instability result for small times in the neighborhood of a general profile ρ 0 ( x ) such that k 0 ( x ) = ρ 0 ′ ( x ) ρ 0 ( x ) is regular enough, bounded, and k 0 ( x ) ( ρ 0 ( x ) ) − 1 2 bounded (which is the case for ρ a ξ ( x L 0 ) ), thanks to the existence of Λ such that γ ≤ Λ for all possible growth rates and at least one growth rate γ belongs to ( Λ 2 , Λ ) . This generalizes the result of Guo and Hwang [Y. Guo, H.J. Hwang, On the dynamical Rayleigh–Taylor instability, Arch. Ration. Mech. Anal. 167 (3) (2003) 235–253], which was obtained in the case ρ 0 ( x ) ≥ ρ l > 0 .