Global Strichartz Estimates for nontrapping Geometries: About an Article by H. Smith and C. Sogge

Generalized Strichartz Estimates on Perturbed Wave Equation and Applications on Strauss Conjecture

In this paper we show a general Strichartz estimate for certain perturbed wave equation, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. Then we give the obstacle version of sharp life span for semilinear wave equations when $n=3,p<p_c$, by using a real interpolation method, and by getting a corresponding finite time Strichartz estimates(see section 3). Finally, as an application of the general Strichartz estimates we have gained, we get the Strauss conjecture for semilinear wave equations…

Global Strichartz Estimates for Solutions to the Wave Equation Exterior to a Convex Obstacle

In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions.

散射流形上正交Strichartz估计的半经典极限

Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds

研究散射流形上正交Strichartz估计的半经典极限，证明非捕获条件下全局时间估计，并展示周期性稳定测地线对估计的影响。

Recent works on the Strauss conjecture

In this review paper, we summarize the current state-of-art on the Strauss conjecture with nontrapping obstacles. Among others, three essential estimates are emphasized and presented: Morawetz-KSS estimates (also known as local energy estimates), weighted Strichartz estimates and generalized Strichartz estimates.

非正曲率流形上的非齐次Strichartz估计及其应用

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。

Strichartz estimates for Dirichlet-wave equations in two dimensions with applications

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors \cite{SmSo00}, Burq \cite{B} and Metcalfe \cite{M} showing that local Strichartz estimates for obstactles imply global ones require that the Sobolev index, $γ$, equal 1/2 when $n=2$. We overcome this difficulty by interpolating between energy estimates ($γ=0$) and ones for $γ=\frac12$ that are generalizations of Minkowski space estimates of Fang and the third autho…

带应用的渐近双曲流形上Klein-Gordon方程的Strichartz估计

Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications

本文建立非陷获渐近双曲流形上Klein-Gordon方程的时间全局Strichartz估计，将其应用于半线性方程的小数据全局存在性与散射研究。

带粗糙衰减势的高阶薛定谔算子的全局Kato光滑性与Strichartz估计

Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials

本文研究带一般有界衰减势的高阶椭圆算子对应的薛定谔型方程，建立了其解的全局Kato光滑性估计与全套Strichartz估计，还引入了带附加导数项的新型Kenig-Ruiz-Sogge型一致Sobolev估计，结果可应用于非线性高阶薛定谔方程的研究。

From semiclassical Strichartz estimates to uniform $L^p$ resolvent estimates on compact manifolds

We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.

Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-pos…

Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials

Let $0<σ<n/2$ and $H=(-Δ)^σ+V(x)$ be Schrödinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2σ}$ with a sharp coupling constant $a\ge -C_{σ,n}$ ($C_{σ,n}$ is the best constant of Hardy's inequality of order $σ$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all …

Time decay for solutions of Schrödinger equations with rough and time dependent potentials

We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schrödinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge …

具有无界势的初始到最终态逆问题与Strichartz估计

The initial-to-final-state inverse problem with unbounded potentials and Strichartz estimates

本文研究了具有无界势的初始到最终态逆问题，通过证明Strichartz估计和Carleman不等式，探讨了数据驱动预测在量子力学中的可行性，并指出Bourgain空间的局限性。

The Strauss conjecture on negatively curved backgrounds

This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $p\in (1, 1+\frac{4}{n-1}]$ for the shifted wave equation on hyperbolic space ${\mathbb H}^n$ involving nonlinearities of the form $\pm |u|^p$ or $\pm|u|^{p-1}u$. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for …