Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Herr, Sebastian

Journées équations aux dérivées partielles (2010), article no. 10, 10 p.

This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^1\left(M\right)$. The results include small data GWP for the quintic NLS in the case of the $3d$ flat rational torus $M={𝕋}^3$ and small data GWP for the corresponding cubic NLS in the cases $M={ℝ}^2\times {𝕋}^2$ and $M={ℝ}^3\times 𝕋$. The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well as critical function space theory. All results mentioned above have been obtained in collaboration with D. Tataru and N. Tzvetkov.

Classification : 35Q55
Mots clés : energy critical nonlinear Schrödinger equations, global well-posedness, critical function spaces, Strichartz estimates

[1] Vasily M. Babič, Eigenfunctions which are concentrated in the neighborhood of a closed geodesic, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 9 (1968), 15–63. | MR | Zbl

[2] Jean-Marc Bouclet and Nikolay Tzvetkov, Strichartz estimates for long range perturbations, Amer. J. Math. 129 (2007), no. 6, 1565–1609. | MR | Zbl

[3] Jean Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. | MR | Zbl

[4] —, On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 1–20. | Zbl

[5] Nicolas Burq, Patrick Gérard, and Nikolay Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569–605. | MR | Zbl

[6] —, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187–223. | MR | Zbl

[7] —, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 255–301. | Numdam | MR | Zbl

[8] —, Global solutions for the nonlinear Schrödinger equation on three-dimensional compact manifolds, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 111–129. | MR | Zbl

[9] Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. | MR | Zbl

[10] Hans Christianson, Cutoff resolvent estimates and the semilinear Schrödinger equation, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3513–3520. | MR | Zbl

[11] James Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, and Terence C. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${ℝ}^3$, Ann. of Math. (2) 167 (2008), no. 3, 767–865. | MR | Zbl

[12] Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace [d’après Bourgain]., Séminaire Bourbaki. Volume 1994/95. Exposés 790-804, Société Mathématique de France. Paris: Astérisque. 237, Exp. No.796, 1996, (French), pp. 163–187. | Numdam | MR | Zbl

[13] Martin Hadac, Sebastian Herr, and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré – AN 26 (2009), no. 3, 917–941, Erratum published at http://dx.doi.org/10.1016/j.anihpc.2010.01.006. | Numdam | MR | Zbl

[14] Sebastian Herr, Daniel Tataru, and Nikolay Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in $4d$ and applications, (in preparation).

[15] —, Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in $H^1\left({𝕋}^3\right)$, arXiv:1005.2832 [math.AP], 2010.

[16] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. | MR | Zbl

[17] Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), no. 2, 217–284. | MR | Zbl

[18] —, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 2007 (2007), no. 16, Art. ID rnm053, 36p. | MR | Zbl

[19] Edmund Landau, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Z. 21 (1924), no. 1, 126–132. | MR

[20] Jeremy Marzuola, Jason Metcalfe, and Daniel Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations., J. Funct. Anal. 255 (2008), no. 6, 1497–1553. | MR | Zbl

[21] M. F. Pyškina, The asymptotic behavior of eigenfunctions of the Helmholtz equation that are concentrated near a closed geodesic, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 15 (1969), 154–160. | MR | Zbl

[22] James V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 (1976), no. 3, 219–242. | MR | Zbl

[23] —, Approximate eigenfunctions of the Laplacian, J. Differential Geometry 12 (1977), no. 1, 87–100. | MR | Zbl

[24] Luc Robbiano and Claude Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Mém. Soc. Math. Fr. (N.S.) (2005), no. 101-102, vi+208. | Numdam | MR | Zbl

[25] Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337–1372. | MR | Zbl

[26] Hideo Takaoka and Nikolay Tzvetkov, On 2D nonlinear Schrödinger equations with data on $ℝ\times 𝕋$, J. Funct. Anal. 182 (2001), no. 2, 427–442. | MR | Zbl

[27] Laurent Thomann, The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces, Bull. Soc. Math. France 136 (2008), no. 2, 167–193. | Numdam | MR | Zbl