Linear instability of $Z$-pinch in plasma (I): Inviscid case

无粘流体中稳定剪切流的不稳定性转变

The transition to instability for stable shear flows in inviscid fluids

该研究针对无粘流体稳定单调剪切流，分析C^s（s<2）扰动下Rayleigh算子的本征值生成，证明存在临界扰动强度m_*区分线性无粘阻尼与不稳定性，并揭示了非线性层面的渐近不稳定性及增长模式。

Compressible, inviscid Rayleigh-Taylor instability

We consider the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. After constructing Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, we turn to an analysis of the equations obtained from linearizing around such a steady state. By a natural variational approach, we construct normal mode solutions that grow exponentially in time with rate like $e^{t \sqrt{\absξ}}$, where $ξ$ is the spatial frequency of the normal mod…

On the Smallness Condition in Linear Inviscid Damping: Monotonicity and Resonance Chains

We consider the linearized Euler equations around a smooth, bilipschitz shear profile $U(y)$ on $\mathbb{T}_L \times \mathbb{R}$. We construct an explicit flow which exhibits linear inviscid damping for $L$ sufficiently small, but for which damping fails if $L$ is large. In particular, similar to the instability results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of an eigenvalue-based argument the underlying mechanism here is shown to be based on a new cascade of resonances moving to higher and higher frequencies in $y$…

二维Taylor-Green涡的无粘不稳定性研究

On the inviscid instability of the 2D Taylor-Green vortex

本文研究二维周期域上Taylor-Green涡的无粘稳定性，数值证明其线性化算子本质谱带内存在不稳定特征值并导致非线性不稳定，同时揭示一种由连续不相关函数族驱动的非模态增长机制，凸显无粘流平衡态的特殊稳定性。

无粘流入-流出问题中的热夫雷不稳定性

Gevrey instability in the inviscid inflow-outflow problem

研究周期通道上二维不可压缩欧拉方程的流入-流出问题，通过设定边界条件，证明在热夫雷空间中该问题全局适定性，指出流入-流出速度边界条件与解的唯一性及解析性的关系。

Stability and instability of Navier boundary layers

We study the inviscid limit problem for the incompressible Navier-Stokes equation on a half-plane with a Navier boundary condition depending on the viscosity. On one hand, we prove the $L^2$ convergence of Leray solutions to the solution of the Euler equation. On the other hand, we show the nonlinear instability of WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^s$ (s>1) norms.

Long time instability of the Couette flow in low Gevrey spaces

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade.

Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime

We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously…

Planar stability critera for viscous shock waves of systems with real viscosity

We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with ``real,'' or partial viscosity. The main result is the establishment of necessary, or ``weak'', and sufficient, or ``strong'', conditions for nonlinear stability analogous to those established by Majda [Ma.1--3] in the inviscid case but (generically) separated by a codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stabil…

Rayleigh-Taylor不稳定性中混合的尺度不变界

Scale invariant bounds for mixing in the Rayleigh-Taylor instability

本文研究了两种可混溶、不可压缩、无粘流体的Rayleigh-Taylor不稳定性，借鉴Saffman-Taylor不稳定性技术，在完全非线性机制下建立了混合区尺寸和内部结构粗化的尺度不变估计，获得了最优标度律并揭示了耗散对减缓混合的强作用。

位力定理及其在两相水波不稳定性中的应用

Virial Theorem and Its Applications in Instability of Two-Phase Water-Wave

本文通过完全非线性系统分析两层无粘无旋流体动力学，建立位力定理并证明下方流体较轻时界面斜率与曲率的多项式增长，即Rayleigh-Taylor不稳定性。

Novel discoveries on the mathematical foundation of linear hydrodynamic stability theory

We present some new discoveries on the mathematical foundation of linear hydrodynamic stability theory. The new discoveries are: 1. Linearized Euler equations fail to provide a linear approximation on inviscid hydrodynamic stability. 2. Eigenvalue instability predicted by high Reynolds number linearized Navier-Stokes equations cannot capture the dominant instability of super fast growth. 3. As equations for directional differentials, Rayleigh equation and Orr-Sommerfeld equation cannot capture the nature of the full differentials.

Spectral stability of inviscid columnar vortices

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particu…

蚂蚁轨迹形成 PDE 模型全局吸引子的存在性与维数下界

Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation

本文研究了蚂蚁轨迹形成非线性 PDE 模型的渐近行为，证明了紧全局吸引子的存在性及维数下界，并分析了齐次稳态在无粘条件下的非线性不稳定性及小参数下的全局渐近稳定性。