论文标题
$ \ mathbf {r}^3 $中不可压缩的粘性雷利 - 泰勒系统的光谱分析
Spectral analysis of the incompressible viscous Rayleigh-Taylor system in $\mathbf{R}^3$
论文作者
论文摘要
层状平滑密度曲线$ρ_0（x_3）$的粘性雷利 - 泰勒模型的线性不稳定性研究量增加了以下4：\ begin {equination} \ bakel} \ label {maineq} {maineq}-λ^2 [ρ_0k^2 [ρ_0k^2 ϕ-ϕ-（ λμ（ϕ^{（4）} -2K^2 ϕ“ + k^4 ϕ）-gk^2ρ_0'dection，\ end {equation}，其中$λ$是时间的增长率，$ k $是波浪数横向到密度的横向。 在$ρ'_0\ geq 0 $紧凑的支持的情况下，我们提供了光谱分析，表明，根据\ cite {hl03}的结果，有一个无限解决方案$（λ_n，ϕ_n）$的无限解决方案序列，带有$λ_n\ rightarrow $ n $ n $ n $ n $ n $ n $ n \ right。 h^4（\ Mathbf {r}）$。在更普遍的情况下，$ρ_0'> 0 $到处都是，$ρ_0$以$ \ pm \ infty $收敛到有限限制$ρ_{\ pm}> 0 $，我们证明存在有限的非琐事solutions $（λ_n，ϕ_n）$。研究线是将两种情况都减少到对紧凑型集合的操作员的研究。
The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile $ρ_0(x_3)$ 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 $λ$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of $ρ'_0\geq 0$ 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 $(λ_n, ϕ_n)$, with $λ_n\rightarrow 0$ when $n\rightarrow +\infty$ and $ϕ_n\in H^4(\mathbf{R})$. In the more general case where $ρ_0'>0$ everywhere and $ρ_0$ converges at $\pm\infty$ to finite limits $ρ_{\pm}>0$, we prove that there exist finitely non trivial solutions $(λ_n, ϕ_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.
