Andrew Comech
We derive damping estimates and asymptotics of $L^p$ operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into $L^2$ almost orthogonality technique of Cotlar-Stein.
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The model example of wave equation with oscillating scale-invariant damping
本文研究了具有振荡尺度不变阻尼的波动方程，发现阻尼系数的振荡会与弹性项产生共振效应，导致部分解的能量衰减率变慢，并通过 Fourier 分析和极坐标变换将问题转化为常微分方程渐近行为与振荡积分的研究。
We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the Hessian determinant of the phase is bounded below. We show in two dimensions that the desired outcome is not always possible, but does occur for a broad class of phases which may be described in terms of the Newton polygon. The estimates obtained are uniform with respect to linear perturbation of the phase and uniform in the cutoff value of the Hessian determinant. In the course of the proof, we investigate …
Singularity Structures of Linear Inviscid Damping in a Channel
针对有限周期渠道中二维 Euler 方程的线性无粘阻尼，本文递归刻画 Green 函数自由与边界部分导致的谱密度奇异性，并揭示流函数近壁奇异性的成因及初始数据、背景流对其正则性的影响。
We find a sharp condition on the density-dependent coefficient of damping of a one-dimensional repulsive Euler-Poisson system, which makes it possible to suppress the formation of singularities in the solution of the Cauchy problem with arbitrary smooth data. In the context of plasma physics, this means the possibility of suppressing the breakdown of arbitrary oscillations of cold plasma.
Oscillatory free boundary problems in stochastic materials
本文研究随机材料中含振荡奇异性的自由边界问题，证明在奇异幂Dini连续且低于阈值时自由边界具有精确正则性，并给出防止其接触高阈值振荡区域的排斥估计。
On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping
本文研究了带非线性阻尼的聚焦质量超临界NLS方程，给出了有限时间爆破的充分条件，证明了阻尼项的非正则化效应，并通过调制分析证明了自相似爆破结构的近似保留。
The effect of singularities and damping on the spectra of photonic crystals
本文研究含奇异介电常数与阻尼的光子晶体，基于Drude-Lorentz模型，分析其对带结构及带隙的影响，并给出了一维显式解及多维积分算子处理方法。
We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation \[ H_εψ:= (-\partial_x^2 + V_0(x) + q(x,x/ε)) ψ= k^2 ψ, \] for $k\in\RR$ and $ε<< 1$. Here, $q(.,y+1)=q(.,y)$, has mean zero and $|V_0(x)+q(x,.)|$ goes to zero as $|x|$ goes to infinity. The distorted plane waves of $H_ε$ are solutions of the form: $e^{\pm ikx}+u^s_\pm(x;k)$, $u^s_\pm$ outgoing as $|x|$ goes to infinity. We derive their $ε$ small asymptotic behavior, from which the asymptotic behavior of scattering…
Limiting absorption principles and linear inviscid damping in the Euler-Boussinesq system in the periodic channel
本文研究周期通道中Euler-Boussinesq系统在线性分层Couette流附近的线性化问题，利用极限吸收原理证明任意正Richardson数下扰动密度和速度场的最优无粘阻尼，并精确刻画了线性化算子的谱结构。
Blow-up criteria and lifespan estimates for semilinear wave equations with time-dependent damping and mass
该研究针对带时变阻尼、质量项及导数型非线性项的半线性波动方程，通过Liouville变换等方法推导了能量解的爆破准则与寿命估计，覆盖质量主导及阻尼扰动抵消的情形。
$E$ Type Singularities for Sufficiently Smooth Functions
本文研究充分光滑函数的Arnol'd型$E_k$（$6\le k\le 8$）奇点，给出其不变性条件与正规形约化结果，利用隐函数定理完成证明，可应用于振荡积分研究。
Landau damping and survival threshold
本文研究了线性化维斯科夫-泊松方程在空间均匀平衡态附近的长时间行为，揭示了波数的生存阈值与 Landau 振荡衰减的关系，证明了振荡波的相速度与粒子速度的共振作用导致衰减。
Nonlinear Landau damping for the unconfined Vlasov-Poisson system in 2D
该论文证明了二维无约束Vlasov-Poisson系统在均匀平衡附近的非线性Landau阻尼，建立了全局存在性、电场衰减估计及散射结果，是首个此类结果。
We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann-Hilbert problems associated with partial differential equations in the Ablowitz-Kaup-Newell-Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author, the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can s…
Well-posedness for the semilinear wave equations with nonlinear damping on manifolds with conical degeneration
本文研究带锥形奇点流形上的半线性波动方程，通过半群方法证局部解存在唯一性，利用势阱证亚临界与临界初始能量下解的整体存在、能量衰减或有限时间爆破，还给出有限时间爆破的充要条件。
A one-dimensional nonlinear model of the so-called upper hybrid oscillations in a magnetoactive plasma is investigated taking into account electron-ion collisions. It is known that both the presence of an external magnetic field of strength $ B_0 $ and a sufficiently large collisional factor $ ν$ help suppress the formation of a finite-dimensional singularity in a solution (breaking of oscillations). Nevertheless, the suppression mechanism is different: an external magnetic field increases the oscillation frequency, and collisions tend to stabilize the medium and suppress oscillations. In term…
In this article, we study the convolution operators $M_k$ with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurfaces $Σ\subset \mathbb{R}^3$ of a hyperbolic equation and smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which exactly one of the principal c…
We study certain families of oscillatory integrals $I_φ(a)$, parametrised by phase functions $φ$ and amplitude functions $a$ globally defined on $\mathbb{R}^d$, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of $I_φ(a)$ are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of $φ$, including elements lying at the boundary of the radial compactification of $\mathbb{R}^d…
In a previous article, \cite{Zill3}, we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution. As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a…
Dispersive decay bounds for the SSH model on the half-line
研究SSH模型的Schrödinger流，利用振荡积分技术证明色散时间衰减估计，量化局部初始条件下能量在晶格中的扩散，并确定衰减率常数对哈密顿量参数的依赖关系。
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