Study of the linear ablation growth rate for the quasi isobaric model of Euler equations with thermal conductivity

In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. The heat flux ~ Q is given by the Fourier law T−ν ~ Q proportional to ∇T , where ν > 1 is the thermal conduction index, and the external force is a gravity field ~g = −g~ex. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The fluid velocity in the denser region is denoted by Va. The system of equations is linearized around a stationary solution, and each perturbed quantity ũ is written using the normal modes method ũ(x, z, t) = R(ū(x, k, γ)e √ ) in order to take into account an increasing solution in time. The resulting linear system is a non self-adjoint fifth order system. Its coefficients depend on x and on physical parameters α, β, α and β being two dimensionless physical constants, given by αβ = kL0 and α β = gL0 V 2 a (introduced in [5]). We study the existence of bounded solutions of this system in the limit α → 0, under the condition β ∈ [β0, 1 β0 ], and the assumption Rγ ∈ [0, 1 β0 ], |γ| ≤ 1 β0 (regime that we studied for a simpler model in [5]) calculating the Evans function Ev(α, β, γ) associated with this linear system. Using rigorous constructions of decreasing at ±∞ solutions of systems of ODE, we prove that, for β ∈ [β0, 1 β0 ], Rγ ∈ [0, 1 β0 ], |γ| ≤ 1 β0 , there exists α1 > 0 such that there is no bounded solution of the linearized system for 0 < α ≤ α1. ∗CEA/DM2S, Centre d’Etudes de Saclay, 91191 Gif sur Yvette Cedex †Université de Paris XIII, LAGA, 93 430 Villetaneuse