Rowan Killip, Monica Visan
We prove scale-invariant Strichartz inequalities for the Schrodinger equation on rectangular tori (rational or irrational) in all dimensions. We use these estimates to give a unified and simpler treatment of local well-posedness of the energy-critical nonlinear Schrodinger equation in dimensions three and four.
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We prove Strichartz estimates over large time scales for the Schrodinger equation set on irrational tori. They are optimal for Lebesgue exponents $p > 6$.
Refinements of Strichartz estimates on tori and applications
本文在环面上证明依赖频率时间区间的三线性与新线性 Strichartz 估计，并将其用于 Airy 传播子，得到修正 KdV 方程解存在性的最优 Sobolev 正则性。
We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.
Strichartz estimates for Schrödinger equations on irrational tori
本文证明了d维无理环面上线性薛定谔方程的新的Schehartz估计。然后，我们利用这些估计证明了非线性薛定谔方程（NLS）在无理环面上的亚临界和临界局部适定性结果。
We prove a bilinear Strichartz type estimate for irrational tori via a decoupling type argument, \cite{bourgain2014proof}, recovering and generalizing the result of \cite{de2006global}. As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two dimensional irrational tori with data of infinite energy.
Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions
在本文中，我们证明了新的多线性Schehartz估计，这是通过使用Bourgain技术得到的。这些估计导致新的临界适定性结果的非线性薛定谔方程的无理环面在二维和三维小的初始数据。在三个维度中，这包括能量临界情况。这延伸了Guo-Oh-Wang最近的工作。
Strichartz Estimates for the Liouville Equation on Euclidean Tori and Applications to Kakeya
本文证明了平坦环面上自由Liouville方程解的时空密度ρ的Strichartz估计，在一维得到最优范围，在高维需加权重，并推测了最优估计范围，部分证明之，最后应用于X射线变换和Kakeya问题。
Lossless Strichartz estimates on rectangular tori over short time intervals
本文研究了在矩形紧致流形上，针对临界指数和端点指数对的无损斯特里茨估计，采用频率局部化初始数据和小时间窗口进行证明。
Sharp Shell-Type Bilinear Decoupling and Improved Strichartz Estimates on Tori
本文在壳型傅里叶支撑限制下建立尖锐双线性解耦不等式，结合多种方法完成证明，并将其应用于推导多种相关估计与结果。
Sharp Strichartz estimates on quadratic irrational tori
本文证明二维周期薛定谔流在积分对角符号下的尖锐L^4 Strichartz估计，揭示算术二分性，并应用于三次双曲NLS的适定性转变。
We provide an asymptotic formula for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi's sharp constant for the linear Strichartz estimate. We calculate the constant in the second term, which differs depending on whether the equation is focussing or defocussing. The sign of this coefficient also changes accordingly.
Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori
我们证明了一般平坦d-环面上任意d.利用这些估计，我们证明了三次非线性薛定谔方程在适当的索伯列夫空间中的局部适定性.在二维和三维，我们证明了多项式的光滑解的Sobolev范数的可能增长的界限。
Strichartz estimates on two-dimensional irrational tori
本文证明二维无理环面上色散 $n_1^2+\alpha n_2^2$ 的 $L^4$ Strichartz估计，损失由傅里叶系数熵控制，改进为对数损失，结合加权关联、光滑局部化与熵递归。
On Strichartz estimates for many-body Schrödinger equation in the periodic setting
本文在周期环面上证明多体薛定谔方程的Strichartz估计，适用于有理和无理环面及小相互作用势，基于Bourgain-Demeter标准估计，是Hong结果的周期类比。
A refinement of the energy method is introduced for dispersive PDE with derivative nonlinearity posed on tori. Key ingredient is a shorttime bilinear Strichartz estimate, which is used in a known combination of perturbative and energy arguments.
We consider the energy critical nonlinear Schrodinger equation on generic irrational tori. Using the long-time Strichartz estimates proved in [8], we establish polynomial upper bounds for higher Sobolev norms for solutions with small energy.
We prove polynomial bounds on the growth of higher Sobolev norms for the nonlinear Schrodinger equation set on a torus, in dimension 3, with super-cubic and sub-quintic nonlinearity. Due to improved Strichartz estimates, these bounds are better for irrational tori than they are for rational tori.
Strichartz estimates for the Schrödinger equation on Zoll manifolds
本文在Zoll流形（含标准球面S^d）上对所有q≥2建立了薛定谔方程解的最优L^q_{t,x}时空估计，核心方法利用拉普拉斯谱的算术性质与双线性振荡积分，将问题化归为一维环面上的Strichartz估计。
Local Well-posedness for 2-D Schrodinger Equation on Irrational Tori and Bounds on Sobolev Norms
本文研究了二维空间中无理环面上的三次薛定谔方程。我们的主要结果是一个改进的非理性环面上的Eschhartz估计。使用这个估计，我们获得了H^s的局部适定性结果，其中s>131/416。我们还得到了高阶Sobolev范数的改进的增长界。
Strichartz type Inequalities for Parabolic and Schrödinger Equations in rearrangement invariant Spaces
本文推广了线性抛物型和薛定谔型偏微分方程初值问题解的经典Schehartz估计在重排不变量（r.i.）空间，并构造一些例子，以证明我们的估计的准确性。
Symmetric hyperbolic Schrödinger equations on tori
本文研究了在周期设定下的对称双曲薛定谔方程，通过改进Bourgain的主要弧方法，证明了在一般流形上的全范围Strichartz估计，并在二维有理流形上建立了两种双曲非线性薛定谔方程的最优局部可解性结果。
Strichartz estimates for the Schrödinger flow on compact Lie groups
本文在紧李群上建立了薛定谔流的尺度不变Strichartz估计，并利用Weyl型和估计及特征标差估计，给出了非矩形环面的完整结果。
Incidence theory and restriction estimates
我们研究了离散约束现象和关联理论之间的相互作用。两个角度进行了探讨。一个是完善的机械开发的托马斯沃尔夫，当结合最近的亚临界估计布尔甘导致新的非理性环面的Eschenhartz估计。另一个连接各种附加能量与Szemerédi-Trotter定理。这两种方法的结合恢复了最著名的经典环面尺寸$n\ge 3$的Eschenhartz估计，没有任何上诉数论。
Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations
本研究针对周期双曲型非线性薛定谔方程，借鉴Killip和Visan方法推导有理与无理环面上的尺度不变Strichartz估计，建立临界空间局部适定性，消除了已有估计中的导数ε损失。
Strichartz estimates for the magnetic Schrödinger equation
我们证明了具有小的不定常磁场的线性磁薛定谔方程的全局尺度不变的Schehartz估计。这是通过构造适当的参数来实现的。作为应用，我们给出了$n\geq 6$维薛定谔映射的一个整体正则性结果.
We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schrödinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials …
Remark on the periodic mass critical nonlinear Schrödinger equation
我们考虑了$\mathbb{T}$和$\mathbb{T}^2$上的质量临界NLS。在$\mathbb{R}^d$的情况下，通过Picard迭代法，我们可以利用Eschhartz估计证明IVP在$L^2$中的适定性（至少对于小数据）。然而，Bourgain（1993）和Takaoka-Tzvetkov（2001）分别给出了$\mathbb{T}$上的$L^6$ Eschhartz和$\mathbb{T}^2$上的$L^4$ Eschhartz的反例，这意味着Eschhartz空间不适合于这些问题的迭代。在本文中，我们给出了一个更强的结果，即即使对于小的初始数据，$\mathbb{T}$和$\mathbb{T}^2$上的IVP在$L^2$中也不能有平滑的数据-解映射。对大多数二维无理环面也得到了同样的结果。
Refined Strichartz estimates and their orthornomal counterparts for Schrödinger equations on torus
本文研究了环面上Schrödinger方程的精细Strichartz估计及其正交 counterparts，揭示了解在混合Lebesgue空间中的更高正则性，并建立了非规范不变非线性项和Hartree方程的适定性。
A Strichartz estimate for de Sitter space
在一类具有C^2度量的渐近de Sitter空间上，利用著名的局部Eschhartz估计和一个重标度论证，证明了共形不变Klein-Gordon方程的Eschhartz估计族.这类度量包括de Sitter空间。我们也给出了这些空间上的半线性Klein-Gordon方程的估计的应用。
Observability estimates for the Schrödinger equation on the equilateral triangle
本文研究等边三角形上 Schrödinger 方程的可观测性，借助 Pinsky 铺砌、奇性传播及带尖锐常数的 Strichartz 估计，在 Neumann 和 Dirichlet 条件下证明无需几何控制条件的新结果。
We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schrödinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} Δu$. Motivated by applications to the mass-critical Schrödinger equation with external potentials (such as the harmonic oscillator) we use a physical space approach.
Angular Regularity and Strichartz Estimates for the Wave Equation
在这里，我们证明了本质上尖锐的线性和双线性的Minkowski空间上的波动方程，其中我们假设的初始数据具有额外的正则性，通常的角动量算子的分数幂的Shehartz型估计。在这种情况下，（q，r）指数的范围大大改善了基于初始数据的平移不变导数和色散不等式的波动方程的可用范围。给出了这个结果的两个证明。
Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary
我们建立了具有边界的黎曼流形$(Ω,\g)$上的薛定谔方程在紧致情形和$Ω$是欧几里得空间中光滑的非捕获障碍物的外部的情形下的薛定谔估计.外部域的估计是尺度不变的范围勒贝格指数$(p,q)$，我们获得这些估计是小于范围内已知的欧几里得空间，但包括关键$L^4_tL^\infty_x$估计，我们用它来给一个简单的证明适定性结果的能量临界薛定谔方程在3维。我们的估计紧流形涉及损失的衍生物的规模不变指标。我们利用这些结果建立了一般紧致流形上某些半线性薛定谔方程的有限能量数据的适定性。
We prove Strichartz estimates for the Schrödinger equation which are scale-invariant up to an $\varepsilon$-loss on products of odd-dimensional spheres. Namely, for any product of odd-dimensional spheres $M=\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$ (so that $M$ is of dimension $d=d_1+\cdots+d_r$ and rank $r$) equipped with rational metrics, the following Strichartz estimate \begin{equation*} \|e^{itΔ}f\|_{L^p(I\times M)}\leq C_\varepsilon\|f\|_{H^{\frac{d}{2}-\frac{d+2}{p}+\varepsilon}(M)} \end{equation*} holds for any $p\geq 2+\frac{8(s-1)}{sr}$, where $$s=\max\left\{\frac{2d_i}{d_i…
Strichartz estimates for the wave equation on flat cones
本文考虑半径为$ρ> 0$的圆上平坦欧氏锥上波动方程的解算子，$ρ> 0$是具有度量$\g(r,θ) = dr^2 + r^2 dθ^2$的流形$\mathbb{R}_+ \times \mathbb{R} / 2 πρ\mathbb{Z}$.使用显式表示的解决方案运营商在相关区域的平面波的传播和衍射的锥点，我们证明色散估计，因此规模不变的波动方程的平锥估计。然后，我们表明，这会产生相应的不等式楔形域，多边形和欧氏曲面与圆锥奇点。这反过来又产生适定性结果的非线性波动方程在这样的流形。Morawetz估计锥也被处理。
In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain \cite{BJFA2005}.
In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invarianc…
We discuss a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs.
Schrödinger equations on compact globally symmetric spaces
本文在紧致整体对称空间上建立尺度不变Strichartz估计及双线性估计，并应用于非线性薛定谔方程的局部适定性。
We prove that there exist Beltrami fields in Euclidean space, with sharp decay at infinity, which have a prescribed set of invariant tori (possibly knotted or linked) that enclose an arbitrarily large number of hyperbolic periodic orbits. These hyperbolic orbits are cablings over the core curve of each torus. Moreover, the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori. We show that an analogous result holds for high-frequency Beltrami fields on the flat 3-torus.
In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain (2005) on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" `a la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows …
Infinite dimensional invariant tori for nonlinear Schrödinger equations
本研究证明无外部参数的圆上非线性薛定谔方程存在大量概周期解，在Kuksin--Poschel 1996年构造的多数有限维KAM环面附近存在无穷维非共振克罗内克环面，回答了哈密顿PDE领域长期存在的问题。
We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, near the edges of the spectral band, coming from rational flow-invariant Lagrangian tori.
We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows to reduce to solving only linear forced equations. We remark that our algorithm works at the…
Invariant tori for the cubic Szegö equation
我们继续研究圆的Hardy空间上的以下哈密顿方程，$$i\partial _tu=Π(|u|^2u)\ ,$$，其中$Π$表示Szegö投影。该方程可以看作是完全非色散发展方程的玩具模型。在以前的工作中，我们证明了这个方程允许一个Lax对，并且它是完全可积的。在本文中，我们构造的作用角变量，这减少了显式的解决方案的方程的对角化问题。因此，我们解决了Hankel算子的逆谱问题。此外，我们建立了相应的不变环面的稳定性。此外，从显式公式中，我们推导出轨道稳定和不稳定行波的分类。
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