Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr

By Nicolas Burq, Patrick Gérard, Nikolay Tzvetkov. 2004

154 citations recorded by the source corpus.

Abstract We study nonlinear Schrodinger equations, posed on a three dimensional Riemannian manifold M. We prove global existence of strong H 1 solutions on M = S 3 and M = S 2 × S 1 as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre, Velo and Bourgain who treated the cases of the Euclidean space R 3 and the torus T 3 = R 3 / Z 3 respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/118ce99e6e31709cf2b828421129d1763bc53c05

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Remarks on the Gibbs measures for nonlinear dispersive equations (2014, background): “ödinger equation on S3. — Let S3 be the unit sphere in R4. We then consider the non linear Schrödinger equation (1.1) ( i∂tu+∆S3u = |u|r−1u, (t,x) ∈ R×S3, u(0,x) = f(x) ∈ Hσ(S3), for 1 ≤ r < 5. In [8] N. Burq, P. Gérard and N. Tzvetkov have shown that (1.1) is globally wellposed in the energy spa”
Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds (2005, background): “Another way is the study of the potential growth of ‖φλ‖Lp(M), see the works by Sogge [18, 19], Sogge-Zelditch [20], the authors [3, 2, 4].”
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].”
On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, methodology): “rticular, the cubic NLS (1.1) is locally well-posed in the (sub-)critical regularity, i.e. s≥ scrit. In the supercritical regime, i.e. s<scrit, on the contrary, (1.1) is known to be ill-posed. See [18, 11, 16, 1]. In the following, however, we consider the Cauchy problem (1.1) with initial data i”
Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, background): “We note that improved Bilinear Strichartz estimates have recently been studied in the case of compact Riemannian manifolds in [22, 23, 59].”
Lp norm estimates of eigenfunctions restricted to submanifolds (2009, background): “(4) 38 (2005), 255–301 [4] Burq N., Gérard P., Tzvetkov N.: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds.”