LINEAR AND NONLINEAR ANALYSIS OF THE VISCOUS 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 gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around an 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 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 $\lambda>0$ 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$k\in L^{-1} mathbb{Z}\setminus\{0\}$, we define a threshold of viscosity coefficient $\mu_c(k,\Xi)$ for the linear instability. So that, in the k-supercritical regime, i.e. $μ>\mu_c(k,\Xi)$, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyễn and then prove that there are infinite nontrivial solutions $(\lambda_n,\phi_n)$ of (0.1)-(0.2) with $\lambda_n\to 0$ as $n\to \infty$ and $\phi_n \in H^4(\mathbf{R}_-) 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 and of Grenier, to prove the nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient