论文标题
具有Navier-Slip边界条件的粘性雷利 - 泰勒系统的线性和非线性分析
Linear and nonlinear analysis of the viscous Rayleigh-Taylor system with Navier-slip boundary conditions
论文作者
论文摘要
在本文中，我们对非线性雷利 - 泰勒对重力驱动的不稳定不稳定感兴趣，可压缩不可压缩的Navier-Stokes方程，在平稳的增长密度概况$ρ_0（x_2）$中，在slab domain $2πl\2πl\ mathbb {t} t} \ times（-1,1,1,1,1 $ l> 0> 0> 0> 0> 0.0 1d曲线）。对粘性雷利 - 泰勒模型的线性不稳定性研究等于对以下颂歌的研究，对有限间隔$（-1,1）$，\ begin {equation} \ label {eqmain}λ^2（ρ_0k^2 ϕ-（ρ_0K^2 ϕ-） ϕ）= gk^2ρ_0'D，\ end {equation}，带有边界条件\ begin {equination} \ label {4thbound} \ begin {cases} ϕ（-1）= 0（1）= 0，\ \ \ \μx） ϕ'（-1），\ end {cases} \ end {equation}其中$λ> 0 $是时间的增长率，$ g> 0 $是重力常数，$ k $是波浪号和两个navier-slip系数$之一$之一对于l^{ - 1} \ Mathbb {z} $中的每个$ k \，我们为线性不稳定性定义了粘度系数$μ_c（k，ξ）$的阈值。因此，在$ k $ -supergalitige的制度中，即$μ>μ_c（k，ξ）$，我们描述了一个光谱分析，以适应由lafitte-nguyen \ cite {ln20}启动的操作员方法\ eqref {eqmain} - \ eqref {4thbound}，带有$λ_n\ to 0 $ as $ n \ to \ 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 $μ> 3 \ sup_ {k \ in l^{ - 1} \ Mathbb {z} \ setMinus \ {0 \}}μ_C（k，k，ξ）$。
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 $ρ_0(x_2)$ in a slab domain $2πL\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ 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 $λ>0$ is the growth rate in time, $g>0$ is the gravity constant, $k$ is the wave number and two Navier-slip coefficients $ξ_{\pm}$ are nonnegative constants. For each $k\in L^{-1}\mathbb{Z}$, we define a threshold of viscosity coefficient $μ_c(k,Ξ)$ for the linear instability. So that, in the $k$-supercritical regime, i.e. $μ>μ_c(k,Ξ)$, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyen \cite{LN20} and prove that there are infinite nontrivial solutions $(λ_n, ϕ_n)_{n\geqslant 1} $ of \eqref{EqMain}-\eqref{4thBound} with $λ_n \to 0$ as $n\to \infty$ and $ϕ_n\in H^4((-1,1))$. 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 $μ>3\sup_{k\in L^{-1}\mathbb{Z}\setminus\{0\}}μ_c(k,Ξ)$.
