On the Cauchy problem for the Hartree type equation in the Wiener algebra

Carles, Rémi; Mouzaoui, Lounès

doi:10.1090/S0002-9939-2014-12072-7

On the Cauchy problem for the Hartree type equation in the Wiener algebra

Authors: Rémi Carles and Lounès Mouzaoui
Journal: Proc. Amer. Math. Soc. 142 (2014), 2469-2482
MSC (2010): Primary 35Q55; Secondary 35A01, 35B30, 35B45, 35B65
DOI: https://doi.org/10.1090/S0002-9939-2014-12072-7
Published electronically: March 17, 2014
MathSciNet review: 3195768
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the mass-subcritical Hartree equation with a homogeneous kernel in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous and is such that its Fourier transform belongs to some Lebesgue space.

[1] Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550
[2] Ioan Bejenaru and Terence Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006), no. 1, 228–259. MR 2204680, https://doi.org/10.1016/j.jfa.2005.08.004
[3] N. Burq, P. Gérard, and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on 𝑆^{𝑑}, Math. Res. Lett. 9 (2002), no. 2-3, 323–335. MR 1909648, https://doi.org/10.4310/MRL.2002.v9.n3.a8
[4] Rémi Carles, Eric Dumas, and Christof Sparber, Multiphase weakly nonlinear geometric optics for Schrödinger equations, SIAM J. Math. Anal. 42 (2010), no. 1, 489–518. MR 2607351, https://doi.org/10.1137/090750871
[5] Rémi Carles, Eric Dumas, and Christof Sparber, Geometric optics and instability for NLS and Davey-Stewartson models, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1885–1921. MR 2984591, https://doi.org/10.4171/JEMS/350
[6] Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047
[7] M. Christ, Power series solution of a nonlinear Schrödinger equation, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 131–155. MR 2333210
[8] M. Christ, J. Colliander, and T. Tao, Instability of the periodic nonlinear Schrödinger equation, archived as http://arxiv.org/abs/math.AP/0311227.
[9] Mathieu Colin and David Lannes, Short pulses approximations in dispersive media, SIAM J. Math. Anal. 41 (2009), no. 2, 708–732. MR 2515782, https://doi.org/10.1137/070711724
[10] James Colliander and Tadahiro Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below 𝐿²(𝕋), Duke Math. J. 161 (2012), no. 3, 367–414. MR 2881226, https://doi.org/10.1215/00127094-1507400
[11] Elena Cordero and Fabio Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr. 281 (2008), no. 1, 25–41. MR 2376466, https://doi.org/10.1002/mana.200610585
[12] Elena Cordero and Davide Zucco, Strichartz estimates for the Schrödinger equation, Cubo 12 (2010), no. 3, 213–239 (English, with English and Spanish summaries). MR 2779383, https://doi.org/10.4067/s0719-06462010000300014
[13] Johannes Giannoulis, Alexander Mielke, and Christof Sparber, High-frequency averaging in semi-classical Hartree-type equations, Asymptot. Anal. 70 (2010), no. 1-2, 87–100. MR 2731651
[14] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582
[15] Ryosuke Hyakuna and Masayoshi Tsutsumi, On existence of global solutions of Schrödinger equations with subcritical nonlinearity for ̂𝐿^{𝑝}-initial data, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3905–3920. MR 2944731, https://doi.org/10.1090/S0002-9939-2012-11314-0
[16] J.-L. Joly, G. Métivier, and J. Rauch, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 167–196 (English, with English and French summaries). MR 1262884
[17] Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
[18] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
[19] David Lannes, Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum, Adv. Differential Equations 6 (2001), no. 6, 731–768. MR 1829094
[20] Luc Molinet, On ill-posedness for the one-dimensional periodic cubic Schrodinger equation, Math. Res. Lett. 16 (2009), no. 1, 111–120. MR 2480565, https://doi.org/10.4310/MRL.2009.v16.n1.a11
[21] Lounès Mouzaoui, High-frequency averaging in the semi-classical singular Hartree equation, Asymptot. Anal. 84 (2013), no. 3-4, 229–245. MR 3136109
[22] Tadahiro Oh and Catherine Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below 𝐿², Kyoto J. Math. 52 (2012), no. 1, 99–115. MR 2892769, https://doi.org/10.1215/21562261-1503772
[23] Jeffrey Rauch, Hyperbolic partial differential equations and geometric optics, Graduate Studies in Mathematics, vol. 133, American Mathematical Society, Providence, RI, 2012. MR 2918544