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Spectral analysis of the incompressible viscous Rayleigh-Taylor system in $\mathbf{R}^3$

The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile $ρ_0(x_3)$ amounts to the study of the following ordinary differential equation of order 4: \begin{equation}\label{MainEq} -λ^2 [ ρ_0 k^2 ϕ- (ρ_0 ϕ')'] = λμ(ϕ^{(4)} - 2k^2 ϕ" + k^4 ϕ) - gk^2 ρ_0'ϕ, \end{equation} where $λ$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of $ρ'_0\geq 0$ compactly supported, we provide a spectral analysis showing that in accordance with the results of \cite{HL03}, there is an infinite sequence of non trivial solutions $(λ_n, ϕ_n)$, with $λ_n\rightarrow 0$ when $n\rightarrow +\infty$ and $ϕ_n\in H^4(\mathbf{R})$. In the more general case where $ρ_0'>0$ everywhere and $ρ_0$ converges at $\pm\infty$ to finite limits $ρ_{\pm}>0$, we prove that there exist finitely non trivial solutions $(λ_n, ϕ_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.