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Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay

Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. (English) Zbl 1116.35109

Ann. Sci. Éc. Norm. Supér. (4) 38, No. 2, 255-301 (2005).
From the abstract: “We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold \(M\). We prove global existence of strong \(H^1\) solutions on \(M=S^3\) and \(M=S^2\times S^1\) 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 \(\mathbb R^3\) and the torus \({\mathbb T}^3=\mathbb R^3/\mathbb Z^3\) respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.” Consider spherical harmonics \(H_p\) of order \(p\) on \(S^d\), where \(d\) is the (arbitrary) dimension of the sphere \(S^d\). The multilinear estimates mentioned above are of the form \[ \| H_pH_q\| _{L^2(S^d)}\leq C_1(d,p,q)\| H_p\| _{L^2(S^d)}\| H_q\| _{L^2(S^d)} \] and \[ \| H_pH_qH_r\| _{L^2(S^d)}\leq C_2(d,p,q,r)\| H_p\| _{L^2(S^d)}\| H_q\| _{L^2(S^d)}\| H_r\| _{L^2(S^d)}, \] where \(C_1(d,p,q)\) and \(C_2(d,p,q,r)\) are given and shown to be asymptotically nearly optimal as \(p,q,r\to\infty\).
Reviewer: Nils Ackermann (México, D.F.)

Cited in 1 Review
Cited in 59 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P05 General topics in linear spectral theory for PDEs
33C55 Spherical harmonics

Keywords:

Nonlinear Schroedinger Equation; Eigenfunction Estimates; Global Existence; Compact Manifold
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References:

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