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 ρ0(x)ρ0(x) such that ρ0(x)=ρaξ(xL0), where ξξ is a nonconstant solution of ξ̇=ξν+1(1−ξ), L0>0L0>0 is the width of the ablation region, ν>1ν>1 is the thermal conductivity exponent, and ρa>0ρa>0 is the maximum density of the fluid. The linearization of the equations around the stationary solution (ρ0,0→,p0), ∇p0=ρ0g→ leads to the study of the Rayleigh equation for the perturbation of the velocity at the wavenumber kk: −ddx(ρ0(x)du¯dx)+k2(ρ0(x)−gγ2ρ0′(x))u¯=0. We denote by the terms ‘eigenmode and growth rate’ an L2(R)L2(R) solution of the Rayleigh equation associated with a value of γγ. The purpose of this paper is twofold:
• derive the following expansion in kL0kL0, for small kL0kL0, of the unique reduced linear growth rate γgk∈[14,1]gkγ2=1+2Γ(1+1ν)(2kL0ν)1ν+a2(kL0)2ν+O(kL0) where a2a2 is explicitly known, provided ν>2ν>2,
• prove the nonlinear instability result for small times in the neighborhood of a general profile ρ0(x)ρ0(x) such that k0(x)=ρ0′(x)ρ0(x) is regular enough, bounded, and k0(x)(ρ0(x))−12 bounded (which is the case for ρaξ(xL0)), thanks to the existence of ΛΛ such that γ≤Λγ≤Λ for all possible growth rates and at least one growth rate γγ belongs to (Λ2,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 ρ0(x)≥ρl>0ρ0(x)≥ρl>0.