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

Abstract

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} -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 $\lambda$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of ${\rho }_0^{\prime }\ge 0$ 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 $({\lambda }_n,{\varphi }_n)$, with ${\lambda }_n\to 0$ when $n\to +\infty$ and ${\varphi }_n\in H^4(\mathbf{R})$. In the more general case where ${\rho }_0^{\prime }>0$ everywhere and ${\rho }_0$ converges at $\pm \infty$ to finite limits ${\rho }_{\pm }>0$, we prove that there exist finitely non trivial solutions $({\lambda }_n,{\varphi }_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.

This paper is revised