Daoyuan Fang, Chengbo Wang
We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of $L^\infty_x$ estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate $L^4_t L^\infty_x$ for $n=2$.
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In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato's smoothing estimate with a singular weight.
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.
We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method consists in applying simple restriction theorems to the level sets of the surfaces studied (for instance, spheres in the case of the cone) and then integrating the inequalities over all level sets. This allows for a different proof of sharp estimates for the wave equation, and of some estimates related to the Euler equations in the rotational framework.
We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing $ L^{\infty}_{x}(\mathbb{R}^{n})$ by $BMO_{x}(\mathbb{R}^{n})$ and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an…
The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schrödinger equation on $\mathbb{R}^n$, $n\ge3$, with the negative inverse-square potential $-σ|x|^{-2}$ in the critical case $σ=(n-2)^2/4$. It turns out that the situation is different from the subcritical case $σ<(n-2)^2/4$ in which the full range of Strichartz estimates is known to be hold. More precisely, splitting the solution into the radial and non-radial parts, we show that (i) the radial part satisfies a weak-type endpoint estimate, which can be regarded as an extension to higher dimensio…
In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schrödinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof of the linear profile decomposition for the Schröindger equation with initial data in the homogeneous Sobolev space; as a consequence, there exists a maximizer for the Sobolev-Strichartz inequality.
The initial value problem for the homogeneous Schrödinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which Strichartz estimates fail.
Strichartz estimates for a time-decaying harmonic oscillator were proven with some assumptions of coefficients for the time-decaying harmonic potentials. The main results of this paper are to remove these assumptions and to enable us to deal with the more general coefficient functions. Moreover, we also prove similar estimates for time-decaying homogeneous magnetic fields.
We prove new inhomogeneous generalized Strichartz estimates, which do not follow from the homogeneous generalized estimates by virtue of the Christ-Kiselev lemma. Instead, we make use of the bilinear interpolation argument worked out by Keel and Tao and refined by Foschi presented in a unified framework. Finally, we give a sample application.
In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.
We compute explicitely the best constants and, by solving some functional equations, we find all maximizers for homogeneous Strichartz estimates for the Schrodinger equation and for the wave equation in the cases when the Lebesgue exponent is an even integer.
In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz estimates.
We look for the optimal range of Lebesque exponents for which inhomogeneous Strichartz estimates are valid. We show that it is larger than the one given by admissible exponents for homogeneous estimates. We prove inhomogeneous estimates adopting the abstract setting and interpolation techniques already used by Keel and Tao for the endpoint case of the homogenenous estimates. Applications to Schrodinger equations are given, which extend previous work by Kato.
The endpoint Strichartz estimates for two-dimensional Schrodinger equations were recovered by averaging the solutions in L^2 in the angular variable by Tao. For Schrodinger equations with defocusing inverse square potential, we proved that the homogeneous endpoint estimates hold under this setting. In particular, the original versions of endpoint estimates hold for radial data.
The endpoint Strichartz estimates for the Schrödinger equation are known to be false in two dimensions. However, if one averages the solution in $L^2$ in the angular variable, we show that the homogeneous endpoint and the retarded half-endpoint estimates hold, but the full retarded endpoint fails. In particular, the original versions of these estimates hold for radial data.
We consider the mass critical NLS on $\mathbb{T}$ and $\mathbb{T}^2$. In the $\mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\mathbb{T}$ and the $L^4$ Strichartz on $\mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\mathbb{T}$ and $\mat…
Strichartz estimates for mixed homogeneous surfaces in three dimensions
本文研究三维混合齐次曲面对应的尖锐混合范数 Strichartz 估计，通过局部分类奇点并结合改进的 Ginibre–Velo 方法与 Ikromov–Müller 新方法，涵盖有无阻尼因子情形并推广了 CKZ13 的结果。
In this note we discuss the question of homogeneous $ L^2 L^{\infty} $ Strichartz estimates for the Wave equation in dimensions $ n \geq 4 $ raised by Fang and Wang and recently shown to fail by Guo, Li, Nakanishi and Yan using probability theory. We record a deterministic example for disproving this estimate.
For Schrödinger equations with a class of slowly decaying repulsive potentials, we show that the solution satisfies global-in-time Strichartz estimates for any admissible pairs. Our admissible class of potentials includes the positive homogeneous potential $Z|x|^{-μ}$ with $Z>0$ and $0<μ<2$ in three and higher dimensions, especially the repulsive Coulomb potential. The proof employs several techniques from scattering theory such as the long time parametrix construction of Isozaki-Kitada type, propagation estimates and local decay estimates.
The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees
该文研究实双曲空间与齐次树上含分数阶 Laplacian 的 Schrödinger 方程，比较其 Strichartz 估计中的导数损失，发现连续双曲空间存在 Knapp 型损失而离散齐次树上因小时间估计平凡而无损失。
We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results in earlier papers of Hassell-Tao-Wunsch and Mizutani which were local in time.
Inhomogeneous Strichartz estimates on manifolds with nonpositive curvature and applications
研究非正曲率紧致流形上薛定谔方程解的非齐次Strichartz估计，通过证明相关估计改进了三次NLS的Sobolev范数增长界，并将齐次Strichartz估计扩展到含临界奇异势的薛定谔算子。
We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schrödinger-type equations we show its close relation with the Schrödinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary …
We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz …
Smoothing and Strichartz estimates for degenerate Schrödinger-type equations
本文研究时间退化Schrödinger型算子的基本估计，通过比较原理推导任意阶算子的整体齐次平滑估计，证明加权Strichartz型估计并应用于半线性Cauchy问题的局部适定性，同时涵盖非退化算子恢复标准结果。
Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations
本文针对非齐次Hartree方程在Sobolev临界指标下初值问题未解决的现状，通过建立所有可能的带奇异权的$L^p$ Strichartz估计，证明了临界情形下解的局部存在性。
Strichartz estimates involving orthonormal systems at the critical summability exponent
本文在临界可求和指数下研究薛定谔算子的正交Strichartz估计，利用限制弱型估计与实插值方法，在ODCA内部证明了新的全局强型估计，补充了已有工作。
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted $L_t^4L_γ^4$ Strichartz norm of the solution on any curve $γ$ in $\mathbb R^2$. For the specific case of a straight line we upgrade this estimate to a weighted Stri…
We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-Δ)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in \cite{BBCRV} concerning weighted homogeneous St…
This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey-Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the …
This article is concerned with the Zakharov-Kuznetsov equation {equation} \label{ZK0} \partial_tu+\partial_xΔu+u\partial_xu=0 . {equation} We prove that the associated initial value problem is locally well-posed in $H^s(\mathbb R^2)$ for $s>\frac12$ and globally well-posed in $H^1(\mathbb R\times \mathbb T)$ and in $H^s(\R^3) $ for $ s>1$. Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of \eqref{ZK0}. In the $\mathbb R^2$ case, we also need to use a recent re…
We observe a link between the window size of mass concentration and the rate of explosion of the Strichartz norm by revisiting Bourgain's mass concentration for the L^2-critical nonlinear Schrodinger equations.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable Lévy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood--Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn-Laplacian.
We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet co…
This paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schrödinger equation posed on a strip domain $\mathbb{R}\times[0,1]$ with non-homogeneous Dirichlet boundary conditions. For any $s\ge0$, if the initial data $φ(x,y)$ is in Sobolev space $H^s(\mathbb{R}\times[0,1])$ and the boundary data $h(x,t)$ is in $$ {\cal H}^s (\mathbb{R} ) = \left \{ h (x, t) \in L^2 ( \mathbb{R}^2 ) \ \big | \ ( 1 + |λ| + |ξ|)^{\frac12} ( 1+ |λ| + |ξ|^2 )^{\frac{s}{2}}\hat h ( λ, ξ) \in L^2 (\mathbb{R}^2 ) \right \} $$ where $\hat h $ is the Fourier transform of $h$ with respect to $t$ and $ x$…
Critical Strichartz estimates for orthonormal systems
本文针对齐次索伯列夫空间中正交系的Strichartz估计公开问题，证明$q=4$且$r<\infty$的临界情形下$\alpha=q/2$时估计成立，方法可推广至分数阶薛定谔传播子。
We study the dispersive properties of the linear Schrödinger equation with a time-dependent potential $V(t,x)$. We show that an appropriate integrability condition in space and time on $V$, i.e. the boundedness of a suitable $L^{r}_{t}L^{s}_{x}$ norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials $V\in L^{r}_tL^{s}_x$.
The Quintic Wave Equation with Kelvin-Voigt Damping: Strichartz estimates, Well-posedness and Global Stabilization
本文研究了五次波方程在凯尔文-沃伊格阻尼下的适定性与稳定性，通过斯特里茨茨估计和唯一连续性性质证明了能量的全局稳定性。
We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain up to some endpoints the full radial Strichartz estimates for the Schrödinger equation. The ideas of proof are based on Shao's ideas \cite{Shao} and some ideas in \cite{GPW} to treat the non-homogeneous case, while at the endpoint we need to use subtle tools to overcome some logarithmic divergence. We also apply the improved Strichartz estimates to the nonlinear problems. First, we prove the small data scattering and large data LWP for the nonlinear Schrödinger equation…
Decay estimates and Strichartz inequalities for a class of dispersive equations on H-type groups
本文研究H型群上由次拉普拉斯算子的光滑函数生成的色散半群衰减估计，借助非交换傅里叶变换等工具通过频率局部化处理非齐次性，推导多类色散方程的新Strichartz不等式并证明时间衰减最优。
Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains
针对三维圆柱域内半经典薛定谔方程，利用相位函数二次结构建立全局非退化性，结合Littlewood-Paley分解，获得带显式导数损失指数的尖锐全局Strichartz估计。
Remarks on the derivation of the virial identity for nonlinear Schrödinger equations
本文改进非线性 Schrödinger 方程位力恒等式的推导，借助解自身性质避免构造近似解序列及权重正则化论证。
Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation
本文利用Lorentz型空间中的正则Strichartz估计，在更弱正则性条件下建立$H^s$-临界非齐次双调和NLS方程的局部适定、小数据全局适定与散射结果。
We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.
We consider the inhomogeneous biharmonic nonlinear Schrödinger equation $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2(\mathbb{R}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $α=\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a st…
In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one- and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic pr…
We show that global Strichartz estimates for magnetic Dirac operators generally fails, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, i. e. with homogeneity-degree more than -1, such that the magnetic field points to a fixed direction, which does not depend on $x\in \mathbb{R}^3$.
We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartz-type estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is ho…
Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schrodinger equation associated to L with a power-like nonlinearity F of degree d. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global well-posedness for small L2 data with no gauge invariance assumption on the nonlinearity F. On the other hand if F is gauge invariant, L2 conservation leads to global well-posedness for arbitrary L2 data. Notice that, in contrast with the Euclidean case, these global well-posedness results hold with no restric…
We first review the $L^2$ bilinear generalizations of the $L^4$ estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions influence the estimates. Moreover, we prove some refinements which are able to simultaneously detect both concentrations and nonconcentrations in Fourier space.
We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^α$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we o…
Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$ and let $Δ_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=Δ_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\ci…
Previous applications of the enclosure method with a finite set of observation data to a mathematical model of electrical impedance tomography are based on the assumption that the conductivity of the background body is homogeneous and known. This paper considers the case when the conductivity is homogeneous and unknown. It is shown that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivi…
Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations
本文研究Benjamin-Ono型Dirac方程，通过构造单位模规范映射推导耦合Schrödinger系统，结合Strichartz估计得到$n\geq3$时$\psi$的先验Strichartz型估计。
Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space
本文研究度量锥上Schrödinger方程的整体时间Strichartz估计，发现允许对范围受算子最小特征值影响，证明了相关猜想，并应用于建立三次非线性Schrödinger方程的适定性与散射理论。
In this note, we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time. So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.
Small and large data scattering for the dispersion-managed NLS
本文研究色散管理非线性薛定谔方程的小数据和大数据散射问题，通过改进的标准方法证明了不同幂次的散射结果，并讨论了聚焦方程的爆破现象。
On the wave equation with Coulomb potential
本文研究 $d\geq 3$ 维排斥性 Coulomb 势波动方程，刻画线性齐次方程解的渐近行为，并借助径向 Strichartz 估计与势能衰减证明散焦情形径向有限能量解全局存在且双向散射。
This article focuses on Chemin's space $\mathcal{S}'_h$ of homogeneous distributions, which was introduced to serve as a basis for realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection $X_h := \mathcal{S}'_h \cap X$ with various Banach spaces $X$, namely supercritical homogeneous Besov spaces and the Lebesgue space $L^\infty$. For each $X$, we find out if the intersection $X_h$ is dense in $X$. If it is not, then we study its closure $C = {\rm clos}(X_h)$ and prove th…
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