Strichartz estimates for non-degenerate Schrödinger equations

We consider the Schrödinger equation with a non-degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schrödinger equation without loss of derivatives including the endpoint case. In contrast to the Riemannian metric case, we need the additional assumptions for the well-posedness of our Schrödinger equation and for proving Strichartz estimates without loss.     

Linear Restriction Estimates for Schrödinger Equation on Metric Cones

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schrödinger operator on metric cone M, where the metric cone M is of the form M = (0, ∞) _r  × Σ, with the cross section Σ being a compact (n ? 1)-dimensional Riemannian manifold (Σ, h) and the equipped metric being g = dr ² + r ² h. Assuming the initial data possesses additional regularity in angular variable θ ∈ Σ, we show some linear restriction estimates for the solutions. In terms of their applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schrödinger equation on two-dimensional metric cones.     

Strichartz Estimates for Schr(?)dinger Equations with Non-degenerate Coefficients

In the present paper,the full range Strichartz estimates for homogeneous Schr(?)dinger equations with non-degenerate and non-smooth coefficients are proved.For inhomogeneous equation,the non-endpoint Strichartz estimates are also obtained.     

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.     

The Schrödinger Equation Type with a Nonelliptic Operator

We consider some linear Schrödinger equation with variable coefficients associated to a smooth symmetric metric g which can be degenerate, without sign and such that g has a submatrix of fixed rank v which is uniformly nondegenerate. In this general setting we prove Strichartz estimates with a loss of derivative on the solution. We also discuss the problem of the control of high frequencies. In particular, we prove that if the equation preserves the H ^s norm for all s ≥ 0, then we obtain almost the same Strichartz estimates as those for the Schrödinger equation associated to a Riemannian metric of dimension 2d ? v.     

A sharpened Strichartz inequality for radial functions

We prove a new sharpened version of the Strichartz inequality for radial solutions of the Schrödinger equation in two dimensions. We establish an improved upper bound for functions that nearly extremize the inequality, with a negative second term that measures the distance from the initial data to Gaussians.     

Endpoint Strichartz estimates for the Klein–Gordon equation in two space dimensions and some applications

We prove the endpoint Strichartz estimates for the Klein–Gordon equation in mixed norms on the polar coordinates in two space dimensions. As an application, similar endpoint estimates for the Schrödinger equation in two space dimensions are shown by using the non-relativistic limit. The existence of global solutions for the cubic nonlinear Klein–Gordon equation in two space dimensions for small data is also shown.     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups

In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.     

Multilinear estimates and well‐posedness for the vortex filament fourth‐order Schrödinger equations

This paper is concerned with the initial value problem for the fourth‐order nonlinear Schrödinger type equation related to the theory of vortex filament. By deriving a fundamental estimate on dyadic blocks for the fourth‐order Schrödinger through the [k,Z]‐multiplier norm method. we establish multilinear estimates for this nonlinear fourth‐order Schrödinger type equation. The local well‐posedness for initial data in with s > 1 ∕ 2 is implied by the multilinear estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

STRICHARTZ ESTIMATES FOR A SCHRÖDINGER OPERATOR WITH NONSMOOTH COEFFICIENTS

ABSTRACT

We prove Strichartz type estimates for the Schrödinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C ² compactly supported perturbation of the identity, satisfying a nontrapping condition.     

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schr?dinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schr?dinger operator. J. Funct. Anal., 274, 605–658(2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader,B.: The cubic fourth-order Schr?dinger equation. J. Funct. Anal., 256, 2473–2517(2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 +8/n p 1 +8/(n-4) in dimensions n ≥ 7.     

Weighted Strichartz Estimates for the Radial Perturbed Schrödinger Equation on the Hyperbolic Space

In this paper, we study the radial Schrödinger equation perturbed with a rough time dependent potential on the hyperbolic space. It is natural to expect that the curvature of the manifold has some influence on the dispersive properties, indeed we obtain the weighted Strichartz estimates for the perturbed Cauchy problem. We shall notice that our weighted Strichartz estimates makes possible to treat the nonlinearity of the form g(Ω, u) which are unbounded as |Ω| → ∞.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Spherically Averaged Endpoint Strichartz Estimates For The Two­Dimensional Schrödinger Equation

The endpoint Strichartz estimates for the Schrödinger equation is known to be false in two dimensions[7]. However, if one averages the solution in L2 in the angular variable, we show that the homogeneous endpoint and the retarded half­endoint estimates hold, but the full retarded endpoint fails. In particular, the original verisions of these estimates hold for radial data     

Remark on energy‐critical NLS in 5D

We reprove global well‐posedness and scattering for the defocusing energy‐critical nonlinear Schrödinger equation in five dimensions. Inspired by the recent work of Killip and Visan, we adapt the Dodson's strategy ‘long‐time Strichartz estimate’ used in the work on mass‐critical nonlinear Schrödinger equation sets. Copyright © 2015 John Wiley & Sons, Ltd.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Lorentz space extension of Strichartz estimates

In this paper, Strichartz estimates for the solution of the Schrödinger evolution equation are considered on a mixed normed space with Lorentz norm with respect to the time variable.

     

Cubic Nonlinear Schrödinger Equation on Three Dimensional Balls with Radial Data

We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of X ^{s, b} spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrödinger flow with particular initial data.     

Strichartz Estimates for Schrödinger Equations on Scattering Manifolds

The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric.