Scattering for solutions of NLS in the exterior of a 2d star--shaped obstacle
Fabrice Planchon, Luis Vega
我们证明了二维非线性薛定谔方程的解在能量空间中是分散的，并且在有界光滑星形障碍物的外部也是如此。非线性势是散焦的，并且至少以五次幂增长。
We prove that solutions to non-linear Schrödinger equations in two dimensions and in the exterior of a bounded and smooth star-shaped obstacle scatter in the energy space. The non-linear potential is defocusing and grows at least as the quintic power.
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Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains
我们证明了3维非俘获域外部散焦三次非线性Schroedinger和Gross-Pitaevskii方程在能量空间中的全局适定性。主要成分是一个半经典的Eschhartz估计与外部域的平滑效果相结合，得到Eschhartz估计。
Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data
我们考虑三维空间中单位球外部的散焦能量临界非线性系统。在索伯列夫空间$\dot H_0^1$中，对Dirichlet边界条件下的初值问题，证明了该方法的全局适定性和大径向初值的离散性.我们还指出，同样的策略可以用来处理球外部的Dirichlet边界条件和径向$\dot H_0^1$初始数据的能量-超临界NLS。
Quintic NLS in the exterior of a strictly convex obstacle
研究三维光滑紧严格凸障碍物外的散焦能量临界非线性薛定谔方程.对于具有Dirichlet边界条件的初值问题，我们证明了在能量空间中对所有初始数据的整体适定性和离散性。
We consider the focusing cubic NLS in the exterior $Ω$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubi…
On 2D NLS on non-trapping exterior domains
二维非线性散焦薛定谔方程的整体存在性和散射性在星形障碍物外的区域和至少以五次幂增长的非线性中是已知的。在本文中，我们推广了所有非陷阱障碍和非线性项的幂严格大于大于四次的整体存在性结果。对于这样的非线性，我们也证明了散射一类所谓的几乎是星形的障碍。
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles
本文研究了星型障碍物外高维拟线性波动方程解的长时间存在性。特别是，我们获得外部域类似物的四维结果的Hörmander的非线性允许依赖于解决方案，而不仅仅是它的第一和第二导数。以前的证明在外部域省略了这种依赖性，因为它没有很好地配合使用的能量方法。
V-shaped transition fronts of monotone bistable reaction-diffusion systems in exterior domains
研究单调双稳反应扩散系统在外部领域中的传播现象，通过构造合适的上下解，证明了V形行波的存在性和单调性，并在完全传播条件下证明其最终恢复V形轮廓。
We prove the scattering for a defocusing nonlinear Schrödinger equation with a sum of two repulsive potentials with strictly convex level surfaces, thus providing a scattering result in a trapped setting similar to the exterior of two strictly convex obstacles.
Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle
本文在假设欧氏空间临界Sobolev空间先验界猜想成立的前提下，证明三维严格凸障碍物外区域上带Dirichlet边界条件的散焦NLS解的全局存在性与散射性，采用集中紧性/刚性论证，通过线性轮廓分解、非线性轮廓嵌入及相关估计排除极小反例。
Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
本文研究二维星形障碍外部拟线性波动方程小数据解的长时间存在性，通过建立Morawetz型能量估计和加权L^2乘积估计，得到几乎尖锐的存在寿命下界。
We study the periodic cubic derivative non-linear Schrödinger equation (dNLS) and the (focussing) quintic non-linear Schrödinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on the line ${\mathbb R}$. We describe the (three parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide a criteria for stabi…
Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities
本文研究了非均匀非线性情况下，$L^2$超临界非线性薛定谔方程在外部领域中归一化山峰解的存在性，通过新型极小-极大论证揭示了非自主非线性在非紧致领域中的新现象。
In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimate…
Asymptotics of the critical non-linear wave equation for a class of non star-shaped obstacles
在3+1维空间中，能量临界非线性波动方程在非俘获障碍物外部区域的散射是已知的星形情况。本文推广了一类非星型障碍物的散射，称之为外部照明。我们使用的主要工具是乘法器的方法，推广的Morawetz乘数，以适应几何形状的障碍物的权重。
We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on th…
Decay estimates for variable coefficient wave equations in exterior domains
本文考虑外部区域上的变系数、不定常波动方程。我们证明了本地化的能量估计，如果该域是星形和全球的时间Ehartz估计，如果该域是严格凸。
In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2<p\leq 8$, while a single mode in…
We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\simeq|u|^{γ-1}u$. We assume that $L$ is a small, long range perturbation of $Δ$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the…
Spherical rigidity for an exterior overdetermined problem with Neumann data prescribed by mean curvature
本文研究了外域中带有由平均曲率给定的Neumann数据的过定椭圆自由边界问题，证明了当Γ≥N-2时，星形域中的球形解具有刚性，且在Γ=N-2时，星形性假设可被去除，刚性在所有有界域中成立。
Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity
针对有界域上带势和非齐次非线性项的NLS方程归一化解问题，通过分析不同参数范围和势的假设，证明了全局极小解、高能解的存在性，部分解决开放问题并给出新结果。
We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is also verified for dimension three or higher. The local energy assumption is satisfied for many important cases, including exterior domain with nontrapping obstacles and flat metric, exterior domain with star-shaped obstacles and small asymptotically Euclidean metric, as well as the nontrapping asymptotically Euclidean manifolds (R^n, g).
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particu…
We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.
The lifespan for 3-dimensional quasilinear wave equations in exterior domains
本文研究了一类具有小初值的拟线性波动方程在外部区域上的长时间解的存在性。允许非线性完全依赖于二次水平的解，而不仅仅是解的一阶和二阶导数。在无边界的情况下，相应的寿命约束是由于林德布拉德，杜和周首先证明了这种长期存在外部的星形障碍。在这里，我们放松的假设的几何形状，只需要有一个足够快的衰减的局部能量的线性齐次波动方程，这允许一些域包含被困射线。关键的一步是证明有用的能量估计涉及的缩放向量场的方法，第二作者和Sogge提供了指导。
We prove symmetry for the p-capacitary potential satisfying $$ Δ_p u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overlineΩ , \; u=1 \, \mbox{ on } Γ, \; \lim_{|x|\rightarrow \infty} u(x)=0 , \; \; \; \; \; \; \; \; 1<p<N, $$ under Serrin's overdetermined condition $$ | \nabla u| = c \mbox{ on } Γ. $$ Here $Ω$ is any bounded domain on which no a priori assumption is made, and $Γ$ denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when $Ω$ is star-shaped. Our proof uses the maximum principle for an appropriate $P$-function, some int…
Almost Global Existence for 4-Dimensional Quasilinear Wave Equations in Exterior Domains
本文研究了四维外区域上具有小初值的拟线性波动方程解的几乎整体存在性。允许非线性依赖于在二次水平的解决方案，以及其第一和第二导数。对于这个问题在无边界设置，Hörmander证明，寿命是有界的以下由$\exp(c/ε)$其中$ε>0$表示的大小柯西数据。后来，杜、第二作者索格和周表明，这种不平等也适用于星形障碍。在三维情况下，我们削弱了对几何的假设，只要求障碍物允许线性齐次波动方程的局部能量足够快地衰减。本文的主要创新之处在于使用了第二作者和Sogge的边界项估计以及Klainerman和Sideris的一个估计的一个变体，该估计将通过杜和周的Sobolev不等式得到。
We consider fractional NLS with focusing power-type nonlinearity $$i \partial_t u = (-Δ)^s u - |u|^{2 σ} u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < σ< \infty$ for $s \geq N/2$ and $0 < σ\leq 2s/(N-2s)$ for $s < N/2$. We prove a general criterion for blowup of radial solutions in $\mathbb{R}^N$ with $N \geq 2$ for $L^2$-supercritical and $L^2$-critical powers $σ\geq 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $Ω\subset \mathbb{R}^N$ in any dimension $N \geq 1$ and subject to exterior Dirichlet conditions. In…
A system of inhomogeneous NLS arising in optical media with a $χ^{(2)}$ nonlinearity, part I : Dynamics
本文研究光学介质中具有 $\chi^{(2)}$ 非线性的非齐次非线性薛定谔方程组，建立向量值 Gagliardo--Nirenberg 不等式，给出全局解存在条件，证明质量超临界下非径向能量散射，并给出爆破解存在判据。
The global behaviors for defocusing wave equations in two dimensional exterior region
本文研究二维外部区域中带星形障碍物和 Dirichlet 边界条件的散焦半线性波动方程，证明解的势能衰减、逐点衰减以及能量空间和临界 Sobolev 空间中的散射，并将 $\mathbb{R}^{1+2}$ 上的相关结论推广到外部区域。
Stability of periodic waves for the fractional KdV and NLS equations
本文针对分数阶KdV和NLS方程，构造了哈密顿量约束极小化的钟形周期波，并严格证明了其非退化性、谱稳定性和轨道稳定性。
General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
针对外区域齐次 $k$-Hessian 方程超定问题，论文结合积分恒等式与几何不等式提出新方法，并建立一般单调公式以证明球特征刻画及相关几何不等式。
We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R} \rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equat…
Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation
本文在三维情形下对2≤k<n的齐次k-Hessian方程外部超定问题，结合参数q与尺度函数，通过不同积分机制建立球刚性结果，将构造从凸域扩展至严格星形域。
Construction of multi-solitary waves solution to the focusing nonlinear Schrödinger equation outside an obstacle in the $L^2$-subcritical case
本文在L^2次临界情形下，针对障碍物外部的聚焦非线性薛定谔方程，利用紧性论证、调制理论与局部能量估计，构造了渐近表现为多个不同速度孤立波之和且满足Dirichlet边界条件的解。
Nonlinear Schrödinger equations near an infinite well potential
本文研究了无量纲非线性薛定谔方程\label{eq:abs1} iΦ t（x，t）= -Δ xΦ+V_ \la（x）Φ+ f（x，Φ），\quad x \in \R^N，\in \R，\tag{$NLS_\la$}的驻波解，其中势$V_\la:\R^N\to\R$接近于无限深势$V_\infty:\R^N\to\R$，i. e.外结构域$\R^N\setminus\Om$上的$V_\infty=\infty$、$V_\infty|_\Om\in L^\infty(\Om)$和$V_\la\to V_\infty$在某种意义上精确地称为$\la\to\infty$。非线性可以是Gross-Pitaevskii类型。具有$\la=\infty$的\eqref{eq:abs1}的解在$\R^N\setminus\Om$上为零且满足Dirichlet边界条件，因此它解\label{eq:abs2} iΦ_t（x，t）&= -Δ_xΦ+V_ \la（x）Φ+ f（x，Φ），& \quad x \in \Om，\ t \in \R Φ（x，t）&= 0 & \quad x \pa \Om，\ t \in \R . \tag{$NLS_\infty$}。我们调查的无限势\eqref{eq:abs2}的解决方案$Φ_\infty$引起附近的解决方案$Φ_\la$的有限势\eqref{eq:abs1…
Inverting covariant exterior derivative
本文提出一种基于庞加莱引理线性同伦算子的算法，用于在星形区域上求逆变协变外导数的逆，以几何和算法方式求解协变常数方程，并讨论了约束条件与解的正则性。
We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument…
Feedback stabilization for entropy solutions of a 2x2 hyperbolic system of conservation laws at a junction
本文研究了星形网络上p-system的反馈镇定问题，通过扩展Coron等人的策略并利用front-tracking算法与指数加权Glimm型Lyapunov泛函控制BV范数，证明了在适当条件下系统的熵解是指数可镇定的。
Hörmander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $\Box u = Q(u, u', u'')$ where $Q$ vanishes to second order and $(\partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $u\partial_αu = \frac{1}{2}\partial_αu^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both re…
Nonlinear Schrödinger equation in an exterior domain
该研究针对非捕获障碍物外区域的三次、五次非线性薛定谔方程，利用局部光滑效应与Strichartz型估计，证明了其在二维、三维中围绕零解及非平凡轨迹的局部精确可控性，并讨论了相关开放问题。
Boundary value problems for Choquard equations
本文证明 Choquard 方程在环形和外部区域上 Neumann 或 Dirichlet 边值问题正径向解的存在性，并给出严格星形域上不存在非平凡正则解的条件。
We consider the cubic nonlinear Schrödinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct modified wave operators for small initial and final data respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on $\mathbb{R}^{D-1}$. In the physical dimension $D=3$, this sy…
We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling toward the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presen…
We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These…
Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials
本文识别了一类使立方非线性薛定谔方程不可积但阻碍遍历性出现的等距能谱捕获势，通过解析和数值方法揭示了在弱非线性与强非线性下功率谱呈准离散尖峰的机制，并指出其在Gross-Pitaevskii方程中的物理相关性。
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