Global existence of solutions to Schrödinger equations on compact riemannian manifolds below H 1
[Existence globale de solutions des équations de Schrödinger sur les variétés riemanniennes compactes en régularité plus faible que H 1 ]
Bulletin de la Société Mathématique de France, Tome 138 (2010) no. 4, pp. 583-613.

Nous nous intéressons dans cet article au caractère bien posé des équations de Schrödinger non-linéaires cubiques défocalisantes sur les variétés riemanniennes compactes sans bord, en régularité H s , s<1, sous certaines conditions bilinéaires de Strichartz. Nous trouvons un s ˜<1 tel que la solution est globale pour s>s ˜.

In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. s<1, under some bilinear Strichartz assumption. We will find some s ˜<1, such that the solution is global for s>s ˜.

DOI : 10.24033/bsmf.2597
Classification : 35Q55, 37K05, 37L50, 81Q20
Keywords: schrödinger equation, compact riemannian manifold, global, I-method
Mot clés : Équation de schrödinger, variété riemanienne compacte, globalité, I-méthode
@article{BSMF_2010__138_4_583_0,
     author = {Zhong, Sijia},
     title = {Global existence of solutions to {Schr\"odinger} equations on compact riemannian manifolds below $H^1$},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {583--613},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {138},
     number = {4},
     year = {2010},
     doi = {10.24033/bsmf.2597},
     mrnumber = {2794885},
     zbl = {1236.35002},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/bsmf.2597/}
}
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Zhong, Sijia. Global existence of solutions to Schrödinger equations on compact riemannian manifolds below $H^1$. Bulletin de la Société Mathématique de France, Tome 138 (2010) no. 4, pp. 583-613. doi : 10.24033/bsmf.2597. http://archive.numdam.org/articles/10.24033/bsmf.2597/

[1] T. Akahori - « Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds », Commun. Pure Appl. Anal. 9 (2010), p. 261-280. | MR | Zbl

[2] R. Anton - « Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains », Bull. Soc. Math. France 136 (2008), p. 27-65. | Numdam | MR | Zbl

[3] M. D. Blair, H. F. Smith & C. D. Sogge - « On Strichartz estimates for Schrödinger operators in compact manifolds with boundary », Proc. Amer. Math. Soc. 136 (2008), p. 247-256. | MR | Zbl

[4] J. Bourgain - « Exponential sums and nonlinear Schrödinger equations », Geom. Funct. Anal. 3 (1993), p. 157-178. | MR | Zbl

[5] -, « Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation », Geom. Funct. Anal. 3 (1993), p. 209-262. | MR | Zbl

[6] -, « Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity », Int. Math. Res. Not. 1998 (1998), p. 253-283. | MR | Zbl

[7] -, « A remark on normal forms and the “I-method” for periodic NLS », J. Anal. Math. 94 (2004), p. 125-157. | MR | Zbl

[8] -, « On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori », in Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, 2007, p. 1-20. | MR | Zbl

[9] N. Burq, P. Gérard & N. Tzvetkov - « Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds », Amer. J. Math. 126 (2004), p. 569-605. | MR | Zbl

[10] -, « Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces », Invent. Math. 159 (2005), p. 187-223. | MR | Zbl

[11] -, « Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations », Ann. Sci. École Norm. Sup. 38 (2005), p. 255-301. | Numdam | MR | Zbl

[12] T. Cazenave - Semilinear Schrödinger equations, Courant Lecture Notes in Math., vol. 10, New York University Courant Institute of Mathematical Sciences, 2003. | Zbl

[13] J. Colliander, M. G. Grillakis & N. Tzirakis - « Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on 2 », Int. Math. Res. Not. 2007 (2007). | Zbl

[14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao - « Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation », Math. Res. Lett. 9 (2002), p. 659-682. | MR | Zbl

[15] -, « Resonant decompositions and the I-method for the cubic nonlinear Schrödinger equation on 2 », Discrete Contin. Dyn. Syst. 21 (2008), p. 665-686. | MR | Zbl

[16] D. De Silva, N. Pavlović, G. Staffilani & N. Tzirakis - « Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D », Discrete Contin. Dyn. Syst. 19 (2007), p. 37-65. | MR

[17] Y. F. Fang & M. G. Grillakis - « On the global existence of rough solutions of the cubic defocusing Schrödinger equation in 𝐑 2+1 », J. Hyperbolic Differ. Equ. 4 (2007), p. 233-257. | MR | Zbl

[18] J. Ginibre - « Le problème de Cauchy pour des équations aux dérivées partielles semi-linéaires périodiques en variables d'espace », Séminaire Bourbaki (1995), exposé no 796. | Numdam | Zbl

[19] R. Killip, T. Tao & M. Visan - « The cubic nonlinear Schrödinger equation in two dimensions with radial data », J. Eur. Math. Soc. (JEMS) 11 (2009), p. 1203-1258. | MR | Zbl

[20] A. Martinez - An introduction to semiclassical and microlocal analysis, Universitext, Springer, 2002. | MR | Zbl

[21] S. Zhong - « The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds », J. Differential Equations 245 (2008), p. 359-376. | MR | Zbl

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