Tadahiro Oh, Oana Pocovnicu
We prove almost sure global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$ with random initial data in $ H^s(\mathbb{R}^3) \times H^{s-1}(\mathbb{R}^3)$ for $s > \frac 12$. The main new ingredient is a uniform probabilistic energy bound for approximating random solutions.
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We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.
In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. We show that a unique solution exists for $u_{0} \in H^{s}(\mathbf{R})$, $s > {8/29}$. This improves the result in [13], which proved global well-posedness for $s > {1/3}$. The main new argument is that we obtain almost Morawetz estimates with improved error.
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
We consider the problem of large data scattering for the quintic nonlinear Schrödinger equation on $\R \times \T^2$. This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a "large scale profile") that controls the asymptotic behavior of the solutions.
We consider the problem of large data scattering for the quintic nonlinear Schrödinger equation on $\R \times \T^2$. This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a "large scale profile") that controls the asymptotic behavior of the solutions.
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We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3 dimensions.
In this article, we prove the scattering for the quintic defocusing nonlinear Schrödinger equation on cylinder $\mathbb{R} \times \mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L^2_x h^α$, $0 < α\le 1$, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^d )$, $d\ge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profi…
We prove that the solution map, associated to the quintic fourth order nonlinear Schrödinger equation, exhibits the norm inflation phenomenon at every point in the Sobolev spaces of super-critical regularity. Indeed, we prove this result separately in the cases of negative and of positive regularity. In the negative regularity case, we prove the result for both the defocusing and focusing equations: in the one dimensional case, the associated solution map exhibits norm inflation in Sobolev spaces of super-critical regularity (including the critical index); in the higher dimensional case, the s…
For the Schrödinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.
We consider the defocusing energy-critical nonlinear Schrödinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.
This second article in a two-part series (following [arXiv:2105.02865], listed here as \cite{L}) proves optimal pointwise decay rates for the quintic defocusing wave equation with large initial data on nonstationary spacetimes, and both the quintic defocusing and quintic focusing wave equations with small initial data on nonstationary spacetimes. We prove a weighted local energy decay estimate, and use local energy decay and Strichartz estimates on these variable-coefficient backgrounds. By using an iteration scheme, we obtain the optimal pointwise bounds. In addition, we explain how the itera…
Stabilization of a perturbed quintic defocusing Schrödinger equation in $\mathbb{R}^{3}$
本文针对$\u211d^{3}$中带阻尼扰动的五次散焦Schr\u00f6dinger方程，利用profile分解与微局部分析证明了其在$H^1$能量水平下的局部指数镇定性，并建立了关键的观测性不等式。
In the article, we prove the large data scattering for two problems, i.e. the defocusing quintic nonlinear Schr{ö}dinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}$ and the defocusing cubic nonlinear Schr{ö}dinger equation on $\mathbb{R}^3$ $\times$ $\mathbb{T}$. Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [36]) on large data scattering for the defocusing critical NL…
In this paper we prove that the defocusing, quintic nonlinear Schrödinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.
Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW
该论文针对实值散焦能量超临界五次波动方程，构造了后向光锥中的离散自相似解，并通过计算机辅助证明其有限时间爆破，首次给出此类方程的实值爆破解。
On the Well-posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on ${\mathbb T}^3$
本文研究了三维环面上三次和五次非线性薛定谔系统的适定性与稳定性，证明了局部适定性和全局适定性，并利用能量-卡西米尔方法证明了稳态解的存在性与稳定性。
We consider the nonlinear Schrödinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper we prove scattering in a larger region where the virial quantity is no longer guaranteed to be sign definite.
We consider the nonlinear Schrödinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The core of this article is the notion of stability of solitary waves. We recall the two standard notions of orbital stability in the context of nonlinear Schrödinger equations, and show that they must be considered as independent from each other. We investigate numerically the notion of orbital stability of ground states in the radially symmetric case, confirming existing conjectures or leading to new ones.
On the global well-posedness for the periodic quintic nonlinear Schrödinger equation
本文证明了在临界Sobolev空间中，若五次散焦非线性薛定谔方程的解在其最大存在区间内有界，则解在环面上全局适定。
The cubic-quintic nonlinear Schrödinger equation with inverse-square potential
本文研究带反平方势的三维三次-五次非线性薛定谔方程，在质量-能量平面位力泛函为正的区域建立散射理论，结果与标准三次-五次NLS的已知散射结论相平行。
Consider the defocusing quintic nonlinear Schrödinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.
We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrodinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic-quintic nonlinear Schrodinger equation with a periodic potential.
We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in \cite{tz}, where global well-posedness was established for any $s>{4/9}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution $Iu$. As a byproduct of our proof we also obtain that…
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_ξ^α$ of sequences of marginal density matrices. The regularity is accounted for by $α>\frac12& if $d=1$, $α>\frac d2-\frac{1}{2(p-1)} $ if $d\geq2$ and $(d,p)\neq(3,2)$, and $α\geq1$ if $(d,p)=(3,2)$, where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $ξ>0$ is arbitrary and determin…
Global well-posedness and scattering for the defocusing $\dot{H}^{\frac{1}{2}}$-critical nonlinear Schrödinger equation in $\mathbb{R}^2$
本文研究 $\mathbb{R}^2$ 中 defocusing quintic 非线性 Schrödinger 方程在临界空间 $\dot{H}^{\frac{1}{2}}$ 中的 Cauchy 初值问题，证明了解的有界性蕴含整体存在性与 scattering。
We prove that the quintic Schrodinger equation with Dirichlet boundary conditions is locally well posed for H^{1}_{0} data on any smooth, non-trapping domain of R^3. The key ingredient is a smoothing effect in L^{5}_{x}L^{2}_{t} for the linear equation. We also derive scattering results for the whole range of defocusing sub-quintic Schrodinger equations outside star-shaped domains.
We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ$$ in three spatial dimensions in the class of solutions with $|ψ(x)|\to c >0$ as $|x|\to\infty$. This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state --- the limiting modulus $c$ corresponds to a local minimum in the energy density. Our arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross--Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geom…
Invariant Gibbs measures for the one-dimensional quintic nonlinear Schrödinger equation in infinite volume
该研究聚焦实数轴上散焦五次非线性薛定谔方程的吉布斯测度不变性问题，依托Bourgain关于三次非线性项的前期工作，采用随机量子化方法证明无限体积Φ^{p+1}_1测度的增长估计，最终完成相关不变性证明。
Threshold dynamics for the 3$d$ radial NLS with combined nonlinearity
本文研究三维径向NLS方程在混合非线性条件下的阈值动力学，重点分析能量临界点处解的行为与分类。
Wellposedness and asymptotic behavior of solutions for the quintic wave equation with nonlocal dissipation
本文研究了带有能量临界非线性项和非局部阻尼机制的半线性波方程，通过构造弱解并利用能量估计和紧致性论证，探讨了非局部阻尼对解的稳定性和衰减行为的影响。
In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} |u|^{p+1} dμdt < C E, \] where $E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $H^{1/2,1/2} \times H^{1/2,-1/2}({\mathbb H}^n)$ if $2 \leq n \leq 6$ and $1<p<p_c = 1+ 4/(n-2)$. As another application we…
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.
Global Well-posedness for the Biharmonic Quintic Nonlinear Schrödinger Equation on $\mathbb{R}^2$
本研究证明了二维五次散焦双调和薛定谔方程的柯西问题在特定索伯列夫空间中全局适定，核心采用$I$-方法构造几乎守恒的修正能量泛函。
Global well-posedness of the energy-critical stochastic nonlinear Schrödinger equation on the three-dimensional torus
本文在三维环面上建立了由加性噪声驱动的散焦能量临界五次随机非线性薛定谔方程的全局适定性，通过原子空间框架与概率性扰动论证，首次在临界空间中解决了周期SNLS的全局适定性问题。
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d\geq1$ dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, \cite{klma}, and used in our earlier work, \cite{chpa2}.
We consider the energy-critical defocusing nonlinear wave equation (NLW) on $\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times H^{s-1}(\mathbb{R}^d)$, with $0< s\leq 1$ if $d=4$, and $0\leq s\leq 1$ if $d=5$. The randomization we consider is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory. Under some additional assumptions, for $d=4$, we also prove the probabilistic continuous dependence of the flow with res…
We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(Ω) \times L^2(Ω)$ for any smooth (compact) domain $Ω\subset \mathbb{R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.
We prove that the defocusing quintic wave equation, with Neumann boundary conditions, is globally wellposed on $H^1_N(Ω) \times L^2(Ω)$ for any smooth (compact) domain $Ω\subset \mathbb{R}^3$. The proof relies on one hand on $L^p$ estimates for the spectral projector by Smith and Sogge, and on the other hand on a precise analysis of the boundary value problem, which turns out to be much more delicate than in the case of Dirichlet boundary conditions.
We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} = \frac{d-2}{2}$. More precisely, given a function on $\mathbb R^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling critical regularity along with small data global existen…
On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement
本文研究带部分谐振子的二维散焦五次非线性薛定谔方程的散射问题，通过建立线性轮廓分解、用新型五次DCR系统逼近非线性轮廓，结合集中紧性/刚性论证证明其在加权Sobolev空间的散射性，并推导相关估计证明质量临界DCR系统的散射理论。
Orbital stability vs. scattering in the cubic-quintic Schrodinger equation
本文研究至多三维聚焦-散焦三次-五次非线性 Schrödinger 方程，分析色散与孤立波轨道稳定性的关系，并给出二维小质量渐近线性及三维稳定性分歧等结论。
We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to define a modification of the energy norm $H^{1}$ that is ``almost conserved'' and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>2/3$ .
We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space $\R^3$ and the flat torus $\T^3$ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this goal by introducing a simultaneous renormalization on the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.
We study low regularity behavior of the nonlinear wave equation in $\mathbb{R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb{R}^2)\times H^{s-1}(\mathbb{R}^2)$, is ill-posed whenever ${0 < s < s_{cr}}$, where the critical exponent $s_{cr}$ depends on the degr…
We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy supercritical wave equations and proved that if the solution is {a priori} bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the $H^2$-critical Sobolev norm for soluti…
We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54…
Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$
本文研究 $S^1$ 上非线性 Schrödinger 方程高 Sobolev 范数的时间增长，通过频率分解建立迭代界，并借助修正能量处理高次非线性及破坏可积性的三次 NLS 变体。
We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.
Improved global well-posedness for mass-critical nonlinear Schrödinger equations on tori
本文证明了环面上质量临界非线性薛定谔方程的新全局适定性结果，涵盖一维和二维情形，并给出了聚焦与散焦情形下的质量条件。
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy subcritical defocusing, and $L^2$ subcritical (de)focusing GP hierarchies, in spaces also used by Erdös, Schlein and Yau in \cite{esy1,esy2}. By use of this tool, we prove a priori $H^1$ bounds for positive semidefinite solutions in those spaces. Moreover, we obtain global well-posedness results for positive semidefinite solutions in the spaces studied in the wo…
In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that for $s<\frac12$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for $s>\frac23$. The same argument can be used to prove that any quintic nonlinear…
We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) …
In this article, we follow the strategies, listed in \cite{Burq2011} and \cite{OhPo}, in dealing with supercritical cubic and quintic wave equations, we obtain that, the equation \begin{equation*} \left\{ \begin{split} &(\partial^2_t-Δ)u+|u|^{p-1}u=0,\ \ 3<p<5 &\big(u,\partial_tu\big)|_{t=0}=(u_0,u_1)\in H^{s}\times H^{s-1}=:\mathcal{H}^s, \end{split} \right. \end{equation*} is almost surely global well-posed in the sense of Burq and Tzvetkov\cite{Burq2011} for any $s\in (\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\fra…
Almost sure pointwise convergence for the 2D periodic quintic NLS
针对二维周期五次非线性薛定谔方程，研究人员证明当数据属于 $s>0$ 的 $H^s(\mathbb T^2)$ 空间时，其概率性逐点收敛到初始数据，该结果改进了确定性情形下 $s<1/3$ 时收敛不成立的结论。
We prove a new smoothing type property for solutions of the 1d quintic Schrödinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.
We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in $\R^{1+3}$, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain and Grillakis, which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain, but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction varia…
Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations
本文通过数值模拟研究了一维分数次立方非聚焦波方程在周期设定下的能量亚临界和超临界情形，发现norm inflation和概率良好性现象。
Revisiting Bourgain's probabilistic construction of solutions to the 2-$d$ cubic NLS
本文重新审视并简化了Bourgain（1996）证明二维环面上defocusing cubic NLS的Gibbs测度不变性时的概率构造论证，核心方法是利用Deng、Nahmod和Yue（2022）提出的random tensor估计。
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