The orthonormal Strichartz inequality on torus

Nakamura, Shohei

doi:10.1090/tran/7982

The orthonormal Strichartz inequality on torus
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Trans. Amer. Math. Soc. 373 (2020), 1455-1476 Request permission

Abstract:

In this paper, motivated by recent works due to Frank-Lewin-Lieb-Seiringer and Frank-Sabin, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in a certain sense. In particular, we will reveal the tradeoff relation between Sobolev regularity and Schatten exponent gain where the $1/p$ derivative-loss Strichartz inequality plays an important role as in the context on compact manifold due to Burq-Gérard-Tzvetkov. An application of the inequality shows the local well-posedness to the periodic Hartree equation describing the infinitely many quantum particles interacting with the power type potential.

References

Jonathan Bennett, Neal Bez, Susana Gutiérrez, and Sanghyuk Lee, On the Strichartz estimates for the kinetic transport equation, Comm. Partial Differential Equations 39 (2014), no. 10, 1821–1826. MR 3250975, DOI 10.1080/03605302.2013.850880
Árpád Bényi and Tadahiro Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen 83 (2013), no. 3, 359–374. MR 3119672, DOI 10.5486/PMD.2013.5529
Neal Bez, Younghun Hong, Sanghyuk Lee, Shohei Nakamura, and Yoshihiro Sawano, On the Strichartz estimates for orthonormal systems of initial data with regularity, Adv. Math. 354 (2019), 106736, 37. MR 3985036, DOI 10.1016/j.aim.2019.106736
J. 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 1209299, DOI 10.1007/BF01896020
Jean Bourgain and Ciprian Demeter, Improved estimates for the discrete Fourier restriction to the higher dimensional sphere, Illinois J. Math. 57 (2013), no. 1, 213–227. MR 3224568
Jean Bourgain and Ciprian Demeter, New bounds for the discrete Fourier restriction to the sphere in 4D and 5D, Int. Math. Res. Not. IMRN 11 (2015), 3150–3184. MR 3373047, DOI 10.1093/imrn/rnu036
Jean Bourgain and Ciprian Demeter, The proof of the $l^2$ decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. MR 3374964, DOI 10.4007/annals.2015.182.1.9
N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569–605. MR 2058384, DOI 10.1353/ajm.2004.0016
N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187–223 (English, with English and French summaries). MR 2142336, DOI 10.1007/s00222-004-0388-x
Nicolas Burq, Patrick Gérard, and Nikolay Tzvetkov, 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 (English, with English and French summaries). MR 2144988, DOI 10.1016/j.ansens.2004.11.003
F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), no. 2, 483–491. MR 2600446, DOI 10.3934/cpaa.2010.9.483
Thomas Chen, Younghun Hong, and Nataša Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal. 224 (2017), no. 1, 91–123. MR 3609246, DOI 10.1007/s00205-016-1068-x
Thomas Chen, Younghun Hong, and Nataša Pavlović, On the scattering problem for infinitely many fermions in dimensions $d\ge 3$ at positive temperature, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 2, 393–416. MR 3765547, DOI 10.1016/j.anihpc.2017.05.002
Seckin Demirbas, Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms, Commun. Pure Appl. Anal. 16 (2017), no. 5, 1517–1530. MR 3661788, DOI 10.3934/cpaa.2017072
Rupert L. Frank, Mathieu Lewin, Elliott H. Lieb, and Robert Seiringer, Strichartz inequality for orthonormal functions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1507–1526. MR 3254332, DOI 10.4171/JEMS/467
Rupert L. Frank and Julien Sabin, Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates, Amer. J. Math. 139 (2017), no. 6, 1649–1691. MR 3730931, DOI 10.1353/ajm.2017.0041
Rupert L. Frank and Julien Sabin, The Stein-Tomas inequality in trace ideals, Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2015–2016, Ed. Éc. Polytech., Palaiseau, 2017, pp. Exp. No. XV, 12. MR 3616317
Patrick Gérard and Vittoria Pierfelice, Nonlinear Schrödinger equation on four-dimensional compact manifolds, Bull. Soc. Math. France 138 (2010), no. 1, 119–151 (English, with English and French summaries). MR 2638892, DOI 10.24033/bsmf.2586
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250, DOI 10.1007/BF02099195
Zihua Guo, Tadahiro Oh, and Yuzhao Wang, Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc. (3) 109 (2014), no. 4, 975–1013. MR 3273490, DOI 10.1112/plms/pdu025
Zihua Guo and Lizhong Peng, Endpoint Strichartz estimate for the kinetic transport equation in one dimension, C. R. Math. Acad. Sci. Paris 345 (2007), no. 5, 253–256 (English, with English and French summaries). MR 2353675, DOI 10.1016/j.crma.2007.07.002
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
Mathieu Lewin and Julien Sabin, The Hartree equation for infinitely many particles I. Well-posedness theory, Comm. Math. Phys. 334 (2015), no. 1, 117–170. MR 3304272, DOI 10.1007/s00220-014-2098-6
Mathieu Lewin and Julien Sabin, The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D, Anal. PDE 7 (2014), no. 6, 1339–1363. MR 3270166, DOI 10.2140/apde.2014.7.1339
Elliott H. Lieb, The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 1–49. MR 1014510, DOI 10.1090/S0273-0979-1990-15831-8
E. H. Lieb, W. Thirring, Bound on kinetic energy of fermions which proves stability of matter, Phys. Rev. Lett. 35 (1975), 687–689.
Andrea R. Nahmod, The nonlinear Schrödinger equation on tori: integrating harmonic analysis, geometry, and probability, Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 1, 57–91. MR 3403081, DOI 10.1090/bull/1516
J. Sabin, The Hartree equation for infinite quantum systems, Journées équations aux dérivées partielles, (2014), Exp. No. 8, 18 pp.
Julien Sabin, Littlewood-Paley decomposition of operator densities and application to a new proof of the Lieb-Thirring inequality, Math. Phys. Anal. Geom. 19 (2016), no. 2, Art. 11, 11. MR 3508209, DOI 10.1007/s11040-016-9215-z
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
G. Staffilani, Dispersive equations and their role beyond PDE, http://math.mit.edu /~gigliola/AMS-Bulletin.pdf.
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266
Luis Vega, Restriction theorems and the Schrödinger multiplier on the torus, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990) IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 199–211. MR 1155865, DOI 10.1007/978-1-4612-2898-1_{1}8
Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945, DOI 10.1007/BF01212420
Frank Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), no. 3, 201–222. MR 1030488, DOI 10.4064/sm-93-3-201-222