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On the Schrodinger equation outside strictly convex obstacles
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This paper proved sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary and deduced sharp (classical) Stochastic estimates for both the H^1-critical (quintic) and subcritical (sub-critical) Schroff equation in 3D.Abstract:
We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3Dread more
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Quintic NLS in the exterior of a strictly convex obstacle
TL;DR: In this paper, the authors consider the defocusing energy-critical nonlinear Schrodinger equation in the exterior of a smooth compact strictly convex obstacle in 3D space and prove global well-posedness and scattering for all initial data in the energy space.
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Global-in-time Strichartz estimates on nontrapping, asymptotically conic manifolds
Andrew Hassell,Junyong Zhang +1 more
TL;DR: In this paper, the authors obtained global-in-time Strichartz estimates without loss of derivatives for the Schrodinger equation on a class of nonsmooth asymptotically conic manifolds.
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Strichartz estimates and the nonlinear Schr\"odinger equation on manifolds with boundary
TL;DR: In this paper, Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds with boundary were established for both the compact case and the case that the exterior of a smooth, non-trapping obstacle in Euclidean space is a smooth obstacle.
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An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation
Tohru Ozawa,Nicola Visciglia +1 more
TL;DR: In this article, the authors revise the classical approach by Brezis-Gallouet to prove global well-posedness for nonlinear evolution equations, including the quartic NLS and half-wave NLS.
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Global Well-Posedness of the Cubic Nonlinear Schrödinger Equation on Closed Manifolds
TL;DR: In this paper, the authors consider the defocusing cubic non-linear Schrodinger equation on general closed Riemannian surfaces and extend the range of global well-posedness to.
References
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Heat kernels and spectral theory
TL;DR: In this paper, the authors introduce the concept of Logarithmic Sobolev inequalities and Gaussian bounds on heat kernels, as well as Riemannian manifolds.
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Endpoint Strichartz estimates
Markus Keel,Terence Tao +1 more
TL;DR: In this paper, an abstract Strichartz estimate for the wave equation (in dimension n ≥ 4) and for the Schrodinger equation (n ≥ 3) was proved.
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Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
TL;DR: In this paper, the authors give a complete solution when S is a quadratic surface given by the duality argument for the special case S {(x, y) yZ xz I} and give the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic invariant norm.
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The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s
Thierry Cazenave,F. B. Weissler +1 more
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On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations
TL;DR: In this article, the existence and scattering results for semilinear wave equations with low regularity data were proved and the minimal regularity that is needed to ensure local existence and well-posedness was determined.