Time-global smoothing estimates for a class of dispersive equations with constant coefficients

Smoothing properties of dispersive equations on non-compact symmetric spaces

We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schrödinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of manifolds with nonpositive sectional curvature and exponential volume growth at infinity, e.g., hyperbolic spaces. We achieve it by proving the Stein-Weiss inequality and the resolvent estimate of the corresponding Fourier multiplier, which are of independent interest. Moreover, we extend the comparison principles to symmetric spaces and deduce different type…

Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications

We prove smoothing estimates for Schrödinger equations $i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.

Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases…

Parabolic smoothing effect and local well-posedness of fifth order semilinear dispersive equations on the torus

We consider the Cauchy problem of fifth order dispersive equations on the torus. We assume that the initial data is sufficiently smooth and the nonlinear term is a polynomial depending on $\partial_x^3 u, \partial_x^2 u, \partial_x u$ and $u$. We prove that the local well-posedness holds on $[-T,T]$ when the nonlinear term satisfies a condition and otherwise, the local well-posedness holds with a smoothing effect only on either $[0,T]$ or $[-T,0]$ and nonexistence result holds on the other time interval, which means that the nonlinear term can not be treated as a perturbation of the linear par…

On Sharpness of the Local Kato Smoothing Property of Dispersive Wave Equations

Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations $$ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $$ enjoy the local smoothing property $$ q\in H^s (\R ^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac{m-1}{2}}_{loc} \left (\R^n\right )\Big ) , $$ where $m$ is the order of the pseudo-differential operator $P(D)$. This property, now called local Kato smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sjölin for the linear Schrödinger equation. In this paper, we show that t…

Strichartz and smoothing estimates for dispersive equations with magnetic potentials

We prove global smoothing and Strichartz estimates for the Schroedinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.

高阶色散方程的平滑效应及其在非线性初值问题中的应用

Smoothing effect for higher order dispersive equations and applications to nonlinear initial value problems

研究\(p\geq2\)的\(p\)-演化方程类相关初值问题，通过线性算子\(P\)次主导系数虚部空间衰减假设产生平滑效应，应用于含导数非线性项的非线性问题，得出解的存在唯一性，涵盖KdV型和Kawahara型等方程。

非线性光滑性蕴含非线性色散方程空间解析半径下界的改进

Nonlinear smoothing implies improved lower bounds on the radius of spatial analyticity for nonlinear dispersive equations

本文通过将问题归结为具有特定额外导数分布的非线性光滑估计，为散焦广义KdV方程和奇次纯幂非线性薛定谔方程建立了改进的解析半径下界，得到对任意ε>0成立的 $σ(T)\gtrsim T^{-\frac{1}{2}-ε}$。

Smoothing estimates for non-dispersive equations

This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper, where dispersive equations were treated. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(ξ)\neq0$ for $ξ\not=0$, the global smoothing estimate $$ \|\langle x\rangle^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} φ(x)\|_{L^2(\mathbb R_t\times\mathbb R^n_x)} \leq C\|φ\|_{L^2(\mathbb R^n_x)} \quad {\rm(}s>1/2{\rm)} $$ is well-known, while it is also known to fail for non-dispers…

Smoothing estimates for evolution equations via canonical transforms and comparison

The paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, several comparison principles are introduced to relate different smoothing estimate…

Nonexistence of small doubly periodic solutions for dispersive equations

We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem so that doubly periodic solutions correspond to fixed points of a certain operator. We prove that this operator is locally contracting, for almost every temporal period, if the Duhamel integral associated to the evolution exhibits a weak smoothing property. This implies the nonexistence of nontrivial, small-amplitude time-periodic solutions for almost every…

Resolvent estimates related with a class of dispersive equations

We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers.

高阶非线性色散方程的尖锐局部存在性与非线性光滑化

Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities

本文针对高阶非线性色散方程，证明在更高阶和更高维下可获得更尖锐的局部适定性，并给出线性与非线性解正则性增益的显式界，应用于广义KdV、ZK和NLS方程。

Smoothing properties of inhomogeneous equations via canonical transforms

The paper describes a new approach to global smoothing problems for inhomogeneous dispersive evolution equations based on an idea of canonical transformation. In our previous papers, we introduced such a method to show global smoothing estimates for homogeneous dispersive equations. It is remarkable that this method allows us to carry out a global microlocal reduction of equations to some low dimensional model cases. The purpose of this paper is to pursue the same treatment for inhomogeneous equations. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms…

Virial identity and dispersive estimates for the $n$-dimensional Dirac equation

We extend to general dimension $n\ge1$ the virial identity proved by Boussaid-D'Ancona-Fanelli for the 3D magnetic Dirac equation. As an application we deduce smoothing and Strichartz estimates for an $n$-dimensional Dirac equation perturbed with a magnetic potential.

非齐次径向相位的色散估计：在弱色散方程和水波模型中的应用

Dispersive estimates for nonhomogeneous radial phases: an application to weakly dispersive equations and water wave models

本文针对一维和二维径向非齐次相位线性色散方程建立了 L1→L∞、Strichartz、局部 Kato 光滑化及 Morawetz 型估计，并将其应用于多种水波模型。

Endpoint maximal and smoothing estimates for Schroedinger equations

For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

Dispersive estimates for the Dirac equation in an Aharonov-Bohm field

We prove local smoothing and weighted Strichartz estimates for the Dirac equation with a Aharonov-Bohm potential. The proof relies on an explicit representation of the solution built in terms of spectral projections.

On dispersive blow-ups for the nonlinear Schrödinger equation

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow…

Comparison of estimates for dispersive equations

This paper describes a new comparison principle that can be used for the comparison of space-time estimates for dispersive equations. In particular, results are applied to the global smoothing estimates for several classes of dispersive partial differential equations.

A fourth-order dispersive flow into Kähler manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

Smoothing estimates for variable coefficients Schroedinger equation with electromagnetic potentials

In this paper we develop the classical multiplier technique to prove a virial identity and smoothing estimates (in a perturbative setting) for the electromagnetic variable coefficients Schroedinger equation.

Energy inequalities and dispersive estimates for wave equations with time-dependent coefficients

We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is complemented by a collection of open problems.

Helmholtz and dispersive equations with variable coefficients on exterior domains

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equation $$ -Lu+zu=f, \qquad -Lu:=\nabla^{b}(a(x)\nabla^{b}u)-c(x)u, \qquad \nabla^{b}:=\nabla+ib(x) $$ with fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping. A…

Dispersive estimates for hyperbolic systems with time-dependent coefficients

This paper is devoted to the study of time-dependent hyperbolic systems and the derivation of dispersive estimates for their solutions. It is based on a diagonalisation of the full symbol within adapted symbol classes in order to extract the essential information about representations of solutions. This is combined with a multi-dimensional van der Corput lemma to derive dispersive estimates.

Smoothing properties for the higher order nonlinear Schrödinger equation with constant coefficients

We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schrödinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber.

径向数据下Dirac方程光滑化估计的最优常数

Optimal constants of smoothing estimates for Dirac equations with radial data

针对Dirac方程最优光滑化常数求解难度高、高维仅得非最优上界的问题，本文借助相关方法得到了任意d≥2带径向初值的d维及一维Schrödinger型、Dirac方程的光滑化估计最优常数。

Well-posedness and parabolic smoothing effect for higher order Schrödinger type equations with constant coefficients

We consider the Cauchy problem of a class of higher order Schrödinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.

Weak dispersive estimates for fractional Aharonov-Bohm-Schrödinger groups

We prove local smoothing, local energy decay and weighted Strichartz inequalities for fractional Schrödinger equations with a Aharonov-Bohm magnetic field in 2D. Explicit representations of the flows in terms of spherical expansions of the Hamiltonians are involved in the study. An improvement of the free estimate is proved, when the total flux of the magnetic field through the unit sphere is not an integer.

某些随机色散方程的几乎必然空间衰减和几乎必然非线性平滑

Almost sure spatial decay and almost sure nonlinear smoothing of some stochastic dispersive equations

研究随机mKdV方程和随机三次KdV - Benjamin - Ono方程的几乎必然非线性平滑、空间衰减及一致收敛性。通过证明局部适定性，建立非线性平滑，得出积分项的相关性质，给出特定条件下局部路径解满足的概率结论。

Dispersive Decay for the 1D Klein-Gordon Equation with Variable Coefficient Nonlinearities

We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case when only the cubic coefficients are variable we prove dispersive decay and smoothness of the solution in weighted spaces with the help of quadratic and cubic normal-forms transformations. In the case of cubic interactions these normal forms appear to be novel.

Global parametrices and dispersive estimates for variable coefficient wave equations

In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity.

Dispersive blow up for nonlinear Schroedinger equations revisited

The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schroedinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey-Stewartson and Gross-Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schroedinger equations in spat…

Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution in one direction of time that is infinitely smooth and the problem in not well-posed in the other direction. Under other conditions on the nonlinearity and the data, we prove that the equation behaves like a dispersive one: there exists a unique local solution in both directions of time.

Third order semilinear dispersive equations related to deep water waves

We present local existence theorem of the initial value problem for third order semilinear dispersive partial differential equations in two space dimensions. This type of equations arises in the study of gravity wave of deep water, and cannot be solved by the classical energy method. To solve the initial value problem, we make full use of pseudodifferential operators with nonsmooth coefficients.

Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation

We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H^s(T), s > s_0 where s_0 = -11/6 + \sqrt{61}/6 \thickapprox -0.5316 < -1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the integration formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szegö equation under randomization of initial data.

三维广义 Zakharov-Kuznetsov 方程解的色散爆破

Dispersive blow-up for solutions to three dimensional generalized Zakharov-Kuznetsov equations

本文研究三维广义 Zakharov-Kuznetsov 方程解的色散爆破现象，通过构造光滑初始数据使全局解在有理数时间点失去 $C^{1}$ 正则性，其核心方法是构造线性解并证明非线性平滑估计。

Dispersive homogenized models and coefficient formulas for waves in general periodic media

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length scale $\varepsilon$; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions $u^\varepsilon$ well by a non-dispersive wave equation on fixed time intervals $(0,T)$. Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive eq…

圆柱上弱色散 KP-II 方程的全局结果

Global results for weakly dispersive KP-II equations on the cylinder

本文研究圆柱上弱色散广义 KP-II 方程，利用 Fourier 解耦得到尖锐 Strichartz 估计，证明拟线性问题在 L² 中的全局适定性，并借助 Kato 光滑效应证明小质量解的长时间衰减。

关于KdV和Benjamin型方程的Kato平滑效应

On Kato's smoothing effects for KdV and Benjamin type equations

研究局部与非局部色散相互作用及非线性类型如何影响解的平滑效应，通过推广KdV和Benjamin方程的模型，证明解具有Kato平滑效应并满足正则性传播原理，揭示了高阶色散项决定解的分数正则性局部增益。

变系数三阶算子的平滑效应

Smoothing effect for third order operators with variable coefficients

本文研究了形如 $D_t-A$ 的变系数算子的平滑效应，证明了齐次与非齐次平滑估计，并据此得到了具有导数非线性的 NLIVP 的适定性结果，同时探讨了主象征双特征曲线的非捕获性质。

关于Kawahara方程的色散量子化与分形化

On dispersive quantization and fractalization for the Kawahara equation

本文研究了具界变差初值的Kawahara方程解的二分行为，推导非线性Duhamel解的平滑估计，结合线性解已知结果，为Talbot效应提供数学描述。

Dispersive Estimates for Manifolds with one Trapped Orbit

For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: \int_0^T || ρ_s e^{it(Δ_g-V)} u_0 ||_{H^{1/2 - ε}(M)}^2 dt \leq C_T || u_0 ||_{L^2(M)}^2, where $ρ_s(x) \in \Ci(M)$ satisfies $ρ_s = <\dist_g(x,x_0)>^{-s}$, $s> \half$, and $V \in \Ci(M)$, $0 \leq V \leq C$ satisfies $|\nabla V| \leq C <\dist(x,x_0)>^{-1-δ}$ for some $δ>0$. From the local smoothing estimate, we…

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation. II

In this paper we continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method …

A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces

A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main results, time-local existence and the uniqueness of a solution to the initial value problem is established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the …

Improved variable coefficient square functions and local smoothing of Fourier integral operators

We establish certain square function estimates for a class of oscillatory integral operators with homogeneous phase functions. These results are employed to deduce a refinement of a previous result of Mockenhaupt Seeger and Sogge \cite{MSS-jams} on the local smoothing property for Fourier integral operators, which arise naturally in the study of wave equations on compact Riemannian manifolds. The proof is an adaptation of the bilinear approach of Tao and Vargas \cite{Tao-Vargas-II}, %as well as Garrigós and Seeger \cite{GaSe09}, and based on bilinear oscillatory integral estimates of Lee \cite…

非捕获外部区域中带位势的波动方程与薛定谔方程的色散估计

Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains

本文研究非捕获外部区域中带短程位势的波动方程与薛定谔方程，通过证明预解估计，得到局部能量可积性、光滑效应，并将自由方程的局部 Strichartz 估计推广为全局估计。

Weak Dispersive estimates for Schrödinger equations with long range potentials

We prove some local smoothing estimates for the Schrödinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay $(1+|x|)^{-γ}$ for some $γ>0$. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.

傅里叶积分算子的平方函数估计与局部平滑

Square function estimates and Local smoothing for Fourier Integral Operators

本文证明了Guth--Wang--Zhang平方函数估计的变系数版本，进而推导出满足电影曲率条件的2+1维傅里叶积分算子的全范围尖锐局部平滑估计，彻底解决了紧致黎曼曲面波方程的局部平滑猜想。

完全离散方案对具有不连续系数的扩散-色散保守律的收敛性

Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

本文研究了一维空间中具有不连续通量函数的扩散-色散超双曲保守律的完全离散方案收敛性，证明了当扩散-色散系数趋于零时，近似解收敛于经典保守律解，并展示了非经典欠压缩激波的存在。

Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients

Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of $L^p-L^q$ norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are identified and investigated. Thus, a comprehensive analysis is carried out for strictly hyperbolic equations of high ord…

粗糙波动方程的局部光滑化

Local smoothing for rough wave equations

该研究针对带$C^r$系数的粗糙波动方程，利用双线性限制估计证明局部光滑化估计，在二维和三维的$C^{1,1}$系数情形恢复了光滑系数下的最优界，并推导出非平凡的$L^p$-$L^p$局部光滑化估计。

正则性的传播与色散爆破现象向高阶广义 KdV 方程的推广

The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations

本文研究了高阶广义 KdV 方程初值问题的光滑性与奇异性，证明了正则性以无穷大速度传播，并通过构造光滑初始数据揭示了由短波或长波聚焦引起的色散爆破现象。