Linear and nonlinear analysis of the viscous Rayleigh-Taylor system with Navier-slip boundary conditions

Abstract

In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile ${\rho }_0(x_2)$ in a slab domain $2\pi L\mathbb{T}\times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ODE on the finite interval $(-1,1)$, \begin{equation}\label{EqMain} 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} with the boundary conditions \begin{equation}\label{4thBound} \begin{cases} ϕ(-1)=ϕ(1)=0,\\ μϕ''(1) = ξ_+ ϕ'(1), \\ μϕ''(-1) =- ξ_- ϕ'(-1), \end{cases} \end{equation} where $\lambda >0$ is the growth rate in time, $g>0$ is the gravity constant, $k$ is the wave number and two Navier-slip coefficients ${\xi }_{\pm }$ are nonnegative constants. For each $k\in L^{-1}\mathbb{Z}$, we define a threshold of viscosity coefficient ${\mu }_c(k,\Xi )$ for the linear instability. So that, in the $k$-supercritical regime, i.e. $\mu >{\mu }_c(k,\Xi )$, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyen \cite{LN20} and prove that there are infinite nontrivial solutions $({\lambda }_n,{\varphi }_n{)}_{n⩾1}$ of \eqref{EqMain}-\eqref{4thBound} with ${\lambda }_n\to 0$ as $n\to \infty$ and ${\varphi }_n\in H^4((-1,1))$. Based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, extending the previous framework of Guo-Strauss \cite{GS95} and of Grenier \cite{Gre00}, to prove the nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely $\mu >3{\sup }_{k\in L^{-1}\mathbb{Z}\setminus \{0\}}{\mu }_c(k,\Xi )$.