no. 4
An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1069-1079.

We revise the classical approach by Brézis–Gallouët to prove global well-posedness for nonlinear evolution equations. In particular we prove global well-posedness for the quartic NLS on general domains Ω in R2 with initial data in H2(Ω)∩H01(Ω), and for the quartic nonlinear half-wave equation on R with initial data in H1(R).

DOI : 10.1016/j.anihpc.2015.03.004
Mots clés : Half-wave equation, Nonlinear Schrödinger equation, Energy estimates, Global existence 
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     title = {An improvement on the {Br\'ezis{\textendash}Gallou\"et} technique for {2D} {NLS} and {1D} half-wave equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1069--1079},
     publisher = {Elsevier},
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Ozawa, Tohru; Visciglia, Nicola. An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1069-1079. doi : 10.1016/j.anihpc.2015.03.004. http://www.numdam.org/articles/10.1016/j.anihpc.2015.03.004/

[1] Anton, R. Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. Fr., Volume 136 (2008), pp. 27–65 | Numdam | MR | Zbl

[2] Bellazzini, J.; Frank, R.; Visciglia, N. Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, Math. Ann., Volume 360 (2014), pp. 653–673 | DOI | MR | Zbl

[3] Blair, M.D.; Smith, H.F.; Sogge, C.D. Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., Volume 354 (2012), pp. 1397–1430 | DOI | MR | Zbl

[4] Brézis, H.; Gallouët, T. Nonlinear Schrödinger evolution equations, Nonlinear Anal., Theory Methods Appl., Volume 4 (1980), pp. 677–681 | DOI | MR | Zbl

[5] Burq, N.; Gérard, P.; Tzvetkov, N. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Math., Volume 126 (2004), pp. 569–605 | DOI | MR | Zbl

[6] Burq, N.; Gérard, P.; Tzvetkov, N. On nonlinear Schrödinger equations in exterior domains, Ann. Inst. Henri Poincaré (2004), pp. 295–318 | Numdam | MR | Zbl

[7] Cazenave, T. Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc., 2003 | MR | Zbl

[8] Elgart, A.; Schlein, B. Mean field dynamics of boson stars, Commun. Pure Appl. Math., Volume 60 (2007), pp. 500–545 | DOI | MR | Zbl

[9] Frank, R.; Lenzmann, E. Uniqueness of non-linear ground states for fractional Laplacians in R , Acta Math., Volume 210 (2013), pp. 261–318 | DOI | MR | Zbl

[10] Fröhlich, J.; Lenzmann, E. Blowup for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1691–1705 | DOI | MR | Zbl

[11] Gérard, P.; Grellier, S. The cubic Szegö equation, Ann. Sci. Éc. Norm. Supér. (4), Volume 43 (2010), pp. 761–810 | Numdam | MR | Zbl

[12] Holmer, J.; Roudenko, S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Commun. Math. Phys., Volume 282 (2008), pp. 435–467 | DOI | MR | Zbl

[13] Ivanovici, O. On the Schrödinger equation outside strictly convex obstacles, Anal. Partial Differ. Equ., Volume 3 (2010), pp. 261–293 | MR | Zbl

[14] Kenig, C.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math., Volume 46 (1993), pp. 527–620 | DOI | MR | Zbl

[15] Kirkpatrick, K.; Lenzmann, E.; Staffilani, G. On the continuum limit for discrete NLS with long-range interactions, Commun. Math. Phys., Volume 317 (2013), pp. 563–591 | DOI | MR | Zbl

[16] Krieger, J.; Lenzmann, E.; Raphael, P. Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal., Volume 209 (2013), pp. 61–129 | DOI | MR | Zbl

[17] Majda, A.J.; McLaughlin, D.W.; Tabak, E.G. A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., Volume 7 (1997), pp. 9–44 | DOI | MR | Zbl

[18] Pocovnicu, O. Explicit formula for the solution of the Szegö equation on the real line and applications, Discrete Contin. Dyn. Syst., Volume 31 (2011), pp. 607–649 | DOI | MR | Zbl

[19] Tsutsumi, M. On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Anal., Theory Methods Appl., Volume 13 (1989), pp. 1051–1056 | DOI | MR | Zbl

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