Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

Abstract

Abstract. We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key L 4 t L ∞ x estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary.

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

“…Burq, Gérard and Tzvetkov could use the Yudovitch-strategy for three dimensional manifolds without boundary due to the regularizing effect of Strichartz estimates. In [4], Blair, Smith and Sogge proved uniqueness of weak solutions of the deterministic NLS on compact 3D manifolds with boundary as an application of their Strichartz estimates on this type of geometry.…”

Scattering in the energy space for the NLS with variable coefficients

“…Note that for exterior domains, Strichartz estimates are known but only locally in time, see e.g. [2], [1] and the references therein. However, research on this topic is advancing rapidly, thus in the general case Ω = R n we decided to assume a priori the validity of Strichartz estimates.…”

Finite dimensionality of the attractor for the hyperbolic Cahn–Hilliard–Oono equation in

“…This result is obtained combining the technique of energy estimates with the classical Strichartz estimates for the linear Schrödinger equation in (see and references therein). Unfortunately, these estimates do not work or are unknown for general bounded domains , so the extension of these results to bounded domains remains an open problem.…”