论文标题
2D Navier-Stokes方程和相关$α$ - 模型的单向流动的不稳定性
Instability of unidirectional flows for the 2D Navier-Stokes equations and related $α$-models
论文作者
论文摘要
我们研究了圆环上线性化的2D Navier-Stokes方程的单向流动的不稳定性。单向流是稳态，其涡度由与单个向量$ \ Mathbf P \ in \ Mathbb z^{2} $相对应的傅立叶模式给出。使用傅立叶系列和几何分解使我们能够分解线性的操作员$ l_ {b} $作用在空间上的$ \ ell^{2}（\ Mathbb z^{2}）$围绕此稳定状态，作为直接的线性操作员的直接总和$ l_ { \ Mathbb z^2 $中的某些矢量$ \ mathbf q \。使用持续分数的方法，我们证明了该稳定状态的线性化运算符$ l_ {b，\ mathbf q} $具有一个特征值，其真实部分则意味着围绕该稳态状态的线性化方程式的指数不稳定。我们进一步获得了$ l_ {b，\ mathbf q} $的不稳定特征值的表征，从扰动驱动决定因素（Fredholm clapitiant）的零表示，与跟踪类操作员$k_λ$相关。我们还将主要的不稳定性结果扩展到Navier-Stokes方程的正则变体（涉及参数$α> 0 $），即二年级流体模型，Navier-Stokes-$ a $ a $和Navier-Stokes-Stokes-Stokes-voigt模型。
We study instability of unidirectional flows for the linearized 2D Navier-Stokes equations on the torus. Unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector $\mathbf p \in \mathbb Z^{2}$. Using Fourier series and a geometric decomposition allows us to decompose the linearized operator $L_{B}$ acting on the space $\ell^{2}(\mathbb Z^{2})$ about this steady state as a direct sum of linear operators $L_{B,\mathbf q}$ acting on $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$. Using the method of continued fractions we prove that the linearized operator $L_{B,\mathbf q}$ about this steady state has an eigenvalue with positive real part thereby implying exponential instability of the linearized equations about this steady state. We further obtain a characterization of unstable eigenvalues of $L_{B,\mathbf q}$ in terms of the zeros of a perturbation determinant (Fredholm determinant) associated with a trace class operator $K_λ$. We also extend our main instability result to cover regularized variants (involving a parameter $α>0$) of the Navier-Stokes equations, namely the second grade fluid model, the Navier-Stokes-$α$ and the Navier-Stokes-Voigt models.
