Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds

By Nicolas Burq, Patrick Gérard, Nikolay Tzvetkov. 2004

546 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8df80fa38c7766e381fe16c0395c742ef350c8a1

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Citations in the corpus

Probabilistic well-posedness for the cubic wave equation (2011, unknown): “Here, such a loss is forbidden and we are going to follow instead a probabilistic version of the strategy initiated by Yudovich [26] to prove global existence for the two-dimensional Euler equation (see also Brezis–Gallouët [7] and Burq–Gérard– Tzvetkov [8] for similar ideas in the context of disper”
Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “See the papers [4, 5, 6, 7, 10] and references therein for further information.”
Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr (2004, methodology): “The nonlinear interaction F satisfies F (0) = 0 and is supposed of the form F = ∂V∂z̄ with V ∈ C∞(C ; R) satisfying (1.2) V (eiθz) = V (z), θ ∈ R, z ∈ C, and, for some α > 1, |∂k1z ∂k2z̄ V (z)| ≤ Ck1,k2(1 + |z|)1+α−k1−k2 .”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings [4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89].”
Strichartz estimates for Schrödinger equations on irrational tori (2013, background): “has been studied extensively in different settings (for example, M = Rd, Td, and certain classes of manifolds) over recent years [16, 39, 28, 13, 3, 4, 9, 10, 30, 21, 22, 20, 42].”
Sharp Strichartz estimates on nontrapping asymptotically conic manifolds (2004, background): “For instance, for admissible (q, r), Burq, Gerard and Tzvetkov [3] showed that the estimate”
Strichartz estimates for long range perturbations (2005, methodology): “In [6] this is done by only invoking L2 considerations and the fact that, on a compact manifold M , L∞(M) is continuously embedded in L2(M).”
Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1 (2008, background): “1) has also been considered in the setting of compact Riemannian manifolds (M, g), see [9, 10, 14, 15].”
An instability property of the nonlinear Schrödinger equation on $S^{d}$ (2002, methodology): “In [4], we obtained a set of Strichartz inequalities as well as local and global well-posedness results for the nonlinear Schrödinger equation (NLS) posed on a compact riemannian manifold (M, g) i∂tu + ∆gu = f(|u|2)u, u(0, x) = u0(x), (1.”
The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates (2016, methodology): “ed to prove global wellposedness for the NLS with Stratonovich noise on compact, two dimensional manifolds, where the dispersive estimate is not valid and has to be replaced by localized version, see [BGT04]. Date: November 22, 2016. 1 2 FABIAN HORNUNG In this article, we want to use the stochastic ”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, methodology): “The equation is locally well-posed in Hs for all s > 0 when M = T2 (Bourgain [6]), in Hs for all s > 1/4 when M = S2 and Hs for all s > 1/2 in the case of a general surface M without boundary (Burq-Gérard-Tzvetkov [10]).”