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
HTML articles powered by AMS MathViewer

by Rémi Carles and Lounès Mouzaoui PDF

Proc. Amer. Math. Soc. 142 (2014), 2469-2482 Request permission

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.

References

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, DOI 10.1007/978-3-642-16830-7
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, DOI 10.1016/j.jfa.2005.08.004
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, DOI 10.4310/MRL.2002.v9.n3.a8
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, DOI 10.1137/090750871
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, DOI 10.4171/JEMS/350
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, DOI 10.1090/cln/010
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
M. Christ, J. Colliander, and T. Tao, Instability of the periodic nonlinear Schrödinger equation, archived as http://arxiv.org/abs/math.AP/0311227.
Mathieu Colin and David Lannes, Short pulses approximations in dispersive media, SIAM J. Math. Anal. 41 (2009), no. 2, 708–732. MR 2515782, DOI 10.1137/070711724
James Colliander and Tadahiro Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\Bbb T)$, Duke Math. J. 161 (2012), no. 3, 367–414. MR 2881226, DOI 10.1215/00127094-1507400
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, DOI 10.1002/mana.200610585
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, DOI 10.4067/s0719-06462010000300014
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
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
Ryosuke Hyakuna and Masayoshi Tsutsumi, On existence of global solutions of Schrödinger equations with subcritical nonlinearity for $\widehat {L}^p$-initial data, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3905–3920. MR 2944731, DOI 10.1090/S0002-9939-2012-11314-0
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
Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
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
Luc Molinet, On ill-posedness for the one-dimensional periodic cubic Schrodinger equation, Math. Res. Lett. 16 (2009), no. 1, 111–120. MR 2480565, DOI 10.4310/MRL.2009.v16.n1.a11
Lounès Mouzaoui, High-frequency averaging in the semi-classical singular Hartree equation, Asymptot. Anal. 84 (2013), no. 3-4, 229–245. MR 3136109
Tadahiro Oh and Catherine Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below $L^2$, Kyoto J. Math. 52 (2012), no. 1, 99–115. MR 2892769, DOI 10.1215/21562261-1503772
Jeffrey Rauch, Hyperbolic partial differential equations and geometric optics, Graduate Studies in Mathematics, vol. 133, American Mathematical Society, Providence, RI, 2012. MR 2918544, DOI 10.1090/gsm/133