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 $\rho_0(x_3)$ amounts to the study of the following ordinary differential equation of order 4: \begin{equation}\label{MainEq} -\lambda^2 [ \rho_0 k^2 \phi - (\rho_0 \phi')'] = \lambda \mu (\phi^{(4)} - 2k^2 \phi" + k^4 \phi) - gk^2 \rho_0'\phi, \end{equation} where $\lambda$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of $\rho'_0\geq 0$ 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 $(\lambda_n, \phi_n)$, with $\lambda_n\rightarrow 0$ when $n\rightarrow +\infty$ and $\phi_n\in H^4(\mathbf{R})$. In the more general case where $\rho_0'>0$ everywhere and $\rho_0$ converges at $\pm\infty$ to finite limits $\rho_{\pm}>0$, we prove that there exist finitely non trivial solutions $(\lambda_n, \phi_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.