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Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. (English) Zbl 1116.35109
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 \(p,q,r\to\infty\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35P05 General topics in linear spectral theory for PDEs
33C55 Spherical harmonics
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References:
[1] Bourgain J. , Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations , Geom. Funct. Anal. 3 ( 1993 ) 107 - 156 . MR 1209299 | Zbl 0787.35097 · Zbl 0787.35097
[2] Bourgain J. , Exponential sums and nonlinear Schrödinger equations , Geom. Funct. Anal. 3 ( 1993 ) 157 - 178 . Article | MR 1209300 | Zbl 0787.35096 · Zbl 0787.35096
[3] Bourgain J. , Eigenfunction bounds for the Laplacian on the n -torus , Internat. Math. Res. Notices 3 ( 1993 ) 61 - 66 . MR 1208826 | Zbl 0779.58039 · Zbl 0779.58039
[4] Bourgain J. , Remarks on Strichartz’ inequalities on irrational tori, Personal communication, 2004.
[5] Burq N. , Gérard P. , Tzvetkov N. , Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds , Amer. J. Math. 126 ( 3 ) ( 2004 ) 569 - 605 . MR 2058384 | Zbl 1067.58027 · Zbl 1067.58027
[6] Burq N. , Gérard P. , Tzvetkov N. , An instability property of the nonlinear Schrödinger equation on \({S}^{d}\) , Math. Res. Lett. 9 ( 2-3 ) ( 2002 ) 323 - 335 . MR 1909648 | Zbl 1003.35113 · Zbl 1003.35113
[7] Burq N. , Gérard P. , Tzvetkov N. , The Cauchy problem for the nonlinear Schrödinger equation on compact manifold , J. Nonlinear Math. Phys. 10 ( 2003 ) 12 - 27 . MR 2063542 · Zbl 1362.35282
[8] Burq N. , Gérard P. , Tzvetkov N. , Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces , Invent. Math. 159 ( 2005 ) 187 - 223 . MR 2142336 | Zbl 1092.35099 · Zbl 1092.35099
[9] Burq N. , Gérard P. , Tzvetkov N. , Multilinear estimates for Laplace spectral projectors on compact manifolds , C. R. Acad. Sci. Paris Ser. I 338 ( 2004 ) 359 - 364 . MR 2057164 | Zbl 1040.58011 · Zbl 1040.58011
[10] Cazenave T. , Semi-Linear Schrödinger Equations , Courant Lecture Notes in Mathematics , vol. 10 , New York University, American Mathematical Society , Providence, RI , 2003 . MR 2002047 | Zbl 1055.35003 · Zbl 1055.35003
[11] Christ M., Colliander J., Tao T. , Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, 2003. arXiv | MR 2018661 · Zbl 1048.35101
[12] Gallot S. , Hulin D. , Lafontaine J. , Riemannian Geometry , Universitext, Springer-Verlag , Berlin , 1990 . MR 1083149 | Zbl 0636.53001 · Zbl 0636.53001
[13] Ginibre J. , Velo G. , On a class of nonlinear Schrödinger equations , J. Funct. Anal. 32 ( 1979 ) 1 - 71 . MR 533219 | Zbl 0396.35029 · Zbl 0396.35029
[14] Ginibre J. , Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain) , Séminaire Bourbaki, Exp. 796, Astérisque 237 ( 1996 ) 163 - 187 . Numdam | MR 1423623 | Zbl 0870.35096 · Zbl 0870.35096
[15] Hörmander L. , The spectral function of an elliptic operator , Acta Math. 121 ( 1968 ) 193 - 218 . MR 609014 | Zbl 0164.13201 · Zbl 0164.13201
[16] Hörmander L. , Oscillatory integrals and multipliers on \(F{L}^{p}\) , Ark. Math. 11 ( 1973 ) 1 - 11 . MR 340924 | Zbl 0254.42010 · Zbl 0254.42010
[17] Kato T. , On nonlinear Schrödinger equations , Ann. Inst. Henri Poincaré Phys. Théor. 46 ( 1987 ) 113 - 129 . Numdam | MR 877998 | Zbl 0632.35038 · Zbl 0632.35038
[18] Klainerman S. , Machedon M. , Remark on Strichartz-type inequalities , Internat. Math. Res. Notices 5 ( 1996 ) 201 - 220 , With appendices by J. Bourgain and D. Tataru. MR 1383755 | Zbl 0853.35062 · Zbl 0853.35062
[19] Klainerman S. , Machedon M. , Finite energy solutions of the Yang-Mills equations in \({R}^{3+1}\) , Ann. of Math. (2) 142 ( 1 ) ( 1995 ) 39 - 119 . MR 1338675 | Zbl 0827.53056 · Zbl 0827.53056
[20] Koch H., Tataru D. , Personal communication, 2004.
[21] Lions J.-L. , Quelques méthodes de résolution des équations aux dérivées partielles non linéaires , Dunod , Paris , 1969 . Zbl 0189.40603 · Zbl 0189.40603
[22] Sogge C. , Oscillatory integrals and spherical harmonics , Duke Math. J. 53 ( 1986 ) 43 - 65 . Article | MR 835795 | Zbl 0636.42018 · Zbl 0636.42018
[23] Sogge C. , Concerning the \({L}^{p}\) norm of spectral clusters for second order elliptic operators on compact manifolds , J. Funct. Anal. 77 ( 1988 ) 123 - 138 . MR 930395 | Zbl 0641.46011 · Zbl 0641.46011
[24] Sogge C. , Fourier Integrals in Classical Analysis , Cambridge Tracts in Mathematics , 1993 . MR 1205579 | Zbl 0783.35001 · Zbl 0783.35001
[25] Stein E.M. , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Monographs in Harmonic Analysis , vol. III , Princeton University Press , Princeton, NJ , 1993 . MR 1232192 | Zbl 0821.42001 · Zbl 0821.42001
[26] Szegö G. , Orthogonal Polynomials , Colloq. Publications , American Mathematical Society , Providence, RI , 1974 . · Zbl 0023.21505
[27] Zygmund A. , On Fourier coefficients and transforms of functions of two variables , Studia Math. 50 ( 1974 ) 189 - 201 . Article | MR 387950 | Zbl 0278.42005 · Zbl 0278.42005
[28] Tao T. , Multilinear weighted convolutions of \({L}^{2}\) functions, and applications to non-linear dispersive equations , Amer. J. Math. 123 ( 2001 ) 839 - 908 . MR 1854113 | Zbl 0998.42005 · Zbl 0998.42005
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