Vittoria Pierfelice

Nonlinear Schrödinger equation on real hyperbolic spaces

Abstract We consider the Schrodinger equation with no radial assumption on real hyperbolic spaces H n . We obtain in all dimensions n ⩾ 2 sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well-posedness results for NLS. Specifically, for small initial data, we prove L 2 and H 1 global well-posedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity F. On the other hand, if F is gauge invariant, L 2 charge is conserved and hence, as in the Euclidean case, it is possible to extend local L 2 solutions to global ones. The corresponding argument in H 1 requires conservation of energy, which holds under the stronger condition that F is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani, for small radial L 2 data or for large radial H 1 data. The second application of our global Strichartz estimates is scattering for NLS both in L 2 and in H 1 , with no radial or gauge invariance assumption. Notice that, on Euclidean spaces R n , this is only possible for the critical power γ = 1 + 4 n and can be false for subcritical powers while, on hyperbolic spaces H n , global existence and scattering of small L 2 solutions hold for all powers 1 γ ⩽ 1 + 4 n . If we restrict to defocusing nonlinearities F, we can extend the H 1 scattering results of Banica, Carles and Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities: the geometry of hyperbolic spaces makes every power-like nonlinearity short range.

Wave and Klein–Gordon equations on hyperbolic spaces

We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator

Schrödinger Equations on Damek–Ricci Spaces

\Delta

Weighted Strichartz estimates for the Schrödinger and wave equations on Damek–Ricci spaces

on real hyperbolic spaces of dimension

The wave equation on hyperbolic spaces

n\!\ge\!2

Some remarks on the Schrödinger equation with a potential in LrtLsx

; as

The Incompressible Navier–Stokes Equations on Non-compact Manifolds

\Delta

The wave equation on Damek–Ricci spaces

has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.

On the Wave Equation Associated to the Hermite and the Twisted Laplacian

In this paper we consider the Laplace–Beltrami operator Δ on Damek–Ricci spaces and derive pointwise estimates for the kernel of e τΔ, when τ ∈ ℂ* with Re τ ≥0. When τ ∈iℝ*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice [4] on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated with a distinguished Laplacian on Damek–Ricci spaces, showing that in this case the standard L 1 → L ∞ estimate fails while suitable weighted Strichartz estimates hold.