Time Strichartz Estimates Research Articles

Strichartz estimates for the Schrödinger equation on non-rectangular two-dimensional tori

We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.

Global endpoint Strichartz estimates for Schrödinger equations on the cylinder [formula omitted

We prove global in time Strichartz estimates on the semi-periodic cylinder with initial data in L2. This extends the local results of Takaoka-Tzvetkov [13] and the global estimates with loss of derivatives of Hani-Pausader [9] and Barron [1].

Local in Time Strichartz Estimates for the Dirac Equation on Spherically Symmetric Spaces

Abstract We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.

Strichartz estimates for non‐degenerate Schrödinger equations

AbstractWe 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.

Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications

We consider the Schrodinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time Strichartz estimates (for Dirichlet boundary conditions), we obtain global Strichartz estimates for initial data in H s , 0 ¤ s ¤ 2 and boundary data in a natural space H s. For s ¥ 1{2, the issue of compatibility conditions requires a thorough analysis of the H s space. As an application we solve nonlinear Schrodinger equations and construct global asymptotically linear solutions for small data. A discussion is included on the appropriate notion of scattering in this framework, and the optimality of the H s space.

Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely e−itΛgσ with σ∈(0,∞)\\{1} and Λg:=−Δg where Δg is the Laplace–Beltrami operator on asymptotically Euclidean manifolds (Rd,g). Let f0∈C0∞(R) be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part (1−f0)(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥2 inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption (1.12), the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part f0(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥3 without using any geometric assumption on g. As a byproduct, we prove global in time Strichartz estimates for the fractional Schrödinger and wave equations on (Rd,g), d≥3 under non-trapping condition.

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on “waveguide” manifolds, such as $\mathbb{R}^2\times \mathbb{T}$ in [X. Cheng et al., On Scattering for the Cubic Defocusing Nonlinear Schrödinger Equation on Waveguide $\mathbb{R}^2 \times \mathbb{T}$, preprint, arXiv:1705.00954 [math.AP], 2017]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in Cheng et al. closed as crucial ingredients for compactness of the critical elements.

Scattering for Nonlinear Schrödinger Equation Under Partial Harmonic Confinement

We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

A local well-posed result for the fifth order KP-II initial value problem

The well-posedness of the real valued fifth order KP-II equation in Sobolev spaces Hs1,s2(R2) has been studied by several authors. It was known to be locally well-posed in Hs,0(R2) for s>−5/4 (see Hadac (2008) [3], Isaza et al. (2006) [5]). We are more interested in the case s=−5/4. We obtain the local well-posedness for s1≥−5/4,s2≥0, which improves previous results by Saut and Tzvetkov (2000) [15] and by Isaza, López and Mejía (2006) [5]. Moreover our main contribution is that we set up the global in time Strichartz estimates for the fifth order KP equation (no matter KP-I or KP-II equation) on dyadic pieces. And especially, on the low frequency part, we obtain a BS→L4 estimate, in which we obtain 3/8 regularity which comes from the effect of ∂x−1∂y2 term.

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.

Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, 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. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).

Strichartz estimates on asymptotically hyperbolic manifolds

We prove local in time Strichartz estimates without loss for the restriction of the solution of the Schroedinger equation, outside a large compact set, on a class of asymptotically hyperbolic manifolds.

Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1

We prove global well-posedness and scattering in H 1 for the defocusing nonlinear Schrödinger equations $$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$ on the hyperbolic spaces $${\mathbb{H}^d}$$ , d ≥ 2, for exponents $${\sigma \in (0, 2/(d-2))}$$ . The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H 1 is not known for any exponent $${\sigma \in (1/d, 2/d]}$$ and is known to fail for $${\sigma \in (0, 1/d]}$$ . Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations

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 Schrodinger 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 inequality in the weighted Lebesgue space due to Muckenhoupt. Under radial symmetry, we get significant gains over the usual HLS inequality and Strichartz estimate.

Strichartz estimates for long range perturbations

We study local in time Strichartz estimates for the Schrödinger equation associated to long range perturbations of the flat Laplacian on the Euclidean space. We prove that in such a geometric situation, outside a large ball centered at the origin, the solutions of the Schrödinger equation enjoy the same Strichartz estimates as in the nonperturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is nontrapping, we prove that the Strichartz estimates hold in the whole space.

A Singular Critical Potential for the Schrödinger Operator

AbstractConsider a real potential V on Rd, d ≥ 2, and the Schrödinger equation:(LS) i∂tu + Δu −Vu = 0, u↾t=0 = u0 ∈ L2.In this paper, we investigate the minimal local regularity of V needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of 1/2 derivative) on solutions of (LS). Prior works show some dispersive properties when V (small at infinity) is in Ld/2 or in spaces just a little larger but with a smallness condition on V (or at least on its negative part).In this work, we prove the critical character of these results by constructing a positive potential V which has compact support, bounded outside 0 and of the order (log |x|)2/|x|2 near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator P := −Δ + V.The elementary construction of V consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator P admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C 2 and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.

Global existence results for nonlinear Schrödinger equations with quadratic potentials

We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.