LINEAR AND NONLINEAR ANALYSIS OF THE RAYLEIGH-TAYLOR SYSTEM WITH NAVIER-SLIP BOUNDARY CONDITIONS

Abstract

In this paper, we are interested in the linear and the nonlinear Rayleigh instability for the incompressible Navier-Stokes equations with Navier-slip boundary conditions around a laminar smooth density profile $\rho_0(x_2)$ being increasing in an infinite slab $2\pi L\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ordinary differential equation on the finite interval $(-1,1)$, \begin{equation}\label{EqMain}-\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} with the boundary conditions \begin{equation}\label{4thBound}\begin{cases}\phi(-1)=\phi(1)=0,\\\mu \phi''(1) = \xi_+ \phi'(1), \\\mu \phi''(-1) =- \xi_- \phi'(-1),\end{cases}\end{equation}where $\lambda$ is the growth rate in time, $k$ is the wave number transverse to the density profile and two Navier-slip coefficients $\xi_{\pm}$ are nonnegative constants. For each $k\in L^{-1}\mathbb{Z}\setminus\{0\}$, we define a threshold of viscosity coefficient $\mu_c(k,\Xi)$ for linear instability. So that, in the $k$-supercritical regime, i.e. $\mu>\mu_c(k,\Xi)$, we provide a spectral analysis adapting the operator method of Lafitte-Nguyễn and then prove that there are infinite solutions of \eqref{EqMain}-\eqref{4thBound}. Secondly, we will extend a result of Grenier, by considering a wider class of initial data to the nonlinear perturbation problem, based on infinitely unstable modes of the linearized problem and we will prove nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely $\mu >3\sup_{k\in L^{-1}\mathbb{Z}\setminus\{0\} }\mu_c(k,\Xi)$