Luis Vega, Nicola Visciglia
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
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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.
We present some a - priori bounds from above and from below for solutions to a class of conformally invariant Schroedinger equations. As a by - product we deduce some new uniqueness results.
We show that the time evolution of the operator $H = -Δ+ i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.
We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.
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 $p$.
This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schroedinger equation has global in time solutions provided that the initial data have the small norm. The key ingredient in t…
In this article we prove time-global smoothing effects of dispersive pseudodifferential equations with constant coefficient radially symmetric symbols on real rank one symmetric spaces of noncompact type. We also discuss gain of regularities according to decay rates of initial values for the Schroedinger evolution equation. We introduce some isometric operators and reduce the arguments to the well-known Euclidean case. In our proof, Helgason's Fourier transform and the Radon transform as an elliptic Fourier integral operator play crucial roles.
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…
We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - Δu + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.
We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the…
We prove the local-in-time well-posedness and the mass and energy conservation laws for a 3d cubic nonlinear Schroedinger equation with a real-valued potential.
We prove some local (in time) wellposedness results for nonlinear Schroedinger equations with rough data, that is, the initial value belongs to some Sobolev space of negative index. The proof uses the Fourier restriction norm method.
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.
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate with a smoothing effect on exterior domains.
Local smoothing estimates for the massless Dirac-coulomb equation in 2 and 3 dimensions
该研究证明了二维及三维下带库仑势的无质量Dirac方程的局部光滑估计，借鉴反平方势相关方程的研究思路，采用分波子空间分解与Dirac-Coulomb算子谱分析的方法完成证明。
We establish local well-posedness of the modified Schroedinger map in H^s, s>3/4.
This paper is a continuation of a previous paper by the same authors, where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian $H_0$ on $\re^n$. In this paper, we consider short-range type perturbations $H$ of the Laplacian on $\re^n$, and we characterize the analytic wave front set of the solution to the Schrödinger equation: $e^{-itH}f$, in terms of that of the free solution: $e^{-itH_0}f$, for $t<0$ in the forward nontrapping region. The same result holds for $t>0$ in the backward nontrapping region. This result is an analytic analogue of results by H…
We prove, by adapting the method of Colliander-Kenig (2002), local well-posedness of the initial-boundary value problem for the one-dimensional nonlinear Schroedinger equation on the half-line under low boundary regularity assumptions.
This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of $n$ intervals ($n\ge 1$). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schrödinger equation with Dirichlet control, the exact controll…
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r'_ξ}, 1/r + 1/r'=1. Similar arguments, combined with a…
We study the existence of stationnary positive solutions for a class of nonlinear Schroedinger equations with a nonnegative continuous potential V. Amongst other results, we prove that if V has a positive local minimum, and if the exponent of the nonlinearity satisfies N/(N-2)<p<(N+2)/(N-2), then for small epsilon the problem admits positive solutions which concentrate as epsilon goes to 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.
It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm.
In this short note, we prove a refinement of bilinear local smoothing estimate to Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.
The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - iΔu = (\nabla \bar{u})^β, |β|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s > s_c := \frac{n}{2} - \frac{1}{m-1}$, $s \ge 0$. In the special case of space dimension $n=1$ a global $L^2$-result is obtained for NLS with the nonlinearity $N(u)= \partial_x (\bar{u} ^2)$. The proof uses the Fourier restriction norm method.
It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed.
This paper has been merged to 0908.3255.
We prove local existence and uniqueness for the Newton-Schroedinger equation in three dimensions. Further we show that the blow-up alternative holds true as well as the continuous dependence of the solution w.r.t. the initial data.
Improved local smoothing estimate for the fractional Schrödinger operator
本文利用Guth的k-broad范数估计，改进了分数阶薛定谔算子（α>1）的局部光滑估计结果。
We analyze in this work the regularity properties of the density operator solu- tion to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schrödinger equation. We show in par- ticular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system when the external potential stays bounded away from zero. Namely, as the adimens…
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
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。
Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities
该研究针对带组合非线性项的双调和薛定谔方程，用新方法证明了归一化基态解的存在性、基态对应限制能量泛函的局部极小值，且驻波会因爆破强不稳定，结果涵盖临界情形。
We prove long time Anderson localization for nonlinear random Schroedinger equation in $\ell^2$ by making a Birkoff normal form type transform to creat an energy barrier where there is essentially no mode propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat "rough" data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.
In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compac…
We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [JŠ14], is a central tool in two of the authors' recent work on quantitative $L^3_x$ blow-up criteria [BP21]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical $L^3_x$ norm must concentrate at scales $\sim \sqrt{T^* - t}$ in the presence of a Type I blow-up.
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…
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…
This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $L^3$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Šverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we …
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.
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…
In this article, we study a $d$-dimensional stochastic quadratic nonlinear Schrödinger equation (SNLS), driven by a fractional derivative (of order $-α<0$) of a space-time white noise: $$\left\{ \begin{array}{l}i\partial_t u-Δu= ρ^2 |u|^2 + \langle \nabla \rangle^{-α}\dot{W} \, , \quad t\in [0,T] \, , \, x\in \mathbb{R}^d \, ,\\ u_0 = ϕ\, ,\end{array}\right.$$ where $ρ:\mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth compactly-supported function. When $α< \frac{d}{2}$, the stochastic convolution is a function of time with values in a negative-order Sobolev space and the model has to be interpr…
We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential enable us to also treat the case of small semi-linear perturbations. In both the linear and the nonlinear instances, we prove that, on average, energy remains bounded and solutions scatter.
We show some new local smoothing estimates of the fractional Schrödinger equations with initial data in $α$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates are sharp in some cases. As applications, the local smoothing estimates could show some new local well-posedness on modulation spaces of the fourth-order nonlinear Schrödinger equations on the line.
We improve local smoothing estimates for fractional Schrödinger equations for $α\in (0,1) \cup (1,\infty)$.
On local smoothing estimates for wave equations
本文研究了波方程在不同维度下的局部平滑估计，通过傅里叶积分算子推导出紧黎曼流形和欧几里得空间中的改进估计。
In this paper we prove that the initial-boundary value problem for the forced non-linear Schroedinger equation with a potential on the half-line is locally and (under stronger conditions) globally well posed, i.e. that there is a unique solution that depends continuously on the force at the boundary and on the initial data. We allow for a large class of unbounded potentials. Actually, for local solutions we have no restriction on the grow at infinity of the positive part of the potential, and for global solutions very mild assumptions that allow, for example, for exponential grow.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spati…
The large time asymptotics of nonlinear multichannel Schroedinger equations
该研究针对带一般空间局域相互作用的非线性多通道薛定谔方程，通过相空间分析与新的外区域先验传播估计，证明径向对称且H¹一致有界的解强渐近于自由波加弱局域化解，可统一分析线性含时势与非线性相互作用。
We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states.
Local smoothing for the Hermite wave equation
本文证明了Hermite波动方程解在$L^p$空间中的局部平滑估计，通过线性化Fourier积分算子将其转化为Klein-Gordon方程问题，并利用$\ell^2$-decoupling估计获得了本质锐利的导数损失结果。
We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying $\ell^{p}$-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.
In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrödinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Recently, Sun, Trélat, Zhang and Zhong have shown it only in spacetime $\mathbb R \times \mathbb R$. Our result resolves this …
We prove a local smoothing result for the Schrödinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm}) depending on the accumulation rate of the critical points of the profile curve. The proof uses an h-dependent version of semiclassical propagation of singularities, and a result on gluing an h-dependent number of cutoff resolvent estimates.
We explore the connection between $k$-broad Fourier restriction estimates and sharp regularity $L^p-L^q$ local smoothing estimates for the solutions of the wave equation in $\mathbb{R}^{n}\times \mathbb{R}$ for all $n \geq 3$ via a Bourgain--Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for $e^{i t \sqrt{-Δ}}$ are not invariant under Lorentz rescaling.
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。
Local smoothing estimates for bilinear Fourier integral operators
提出双线性傅里叶积分算子的局部光滑猜想，证明线性局部光滑猜想蕴含该双线性版本，并建立二维情形下的局部光滑估计，对高维情形给出部分进展。
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schrödinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.
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.
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