Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains

Anton, Ramona

Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 1, pp. 27-65.

Nous considérons le problème de Cauchy pour l'équation de Schrödinger non linéaire sur un domaine du plan avec des conditions aux limites de Dirichlet. Nous prouvons que le problème est bien posé et qu'il existe une solution globale pour une non linéarité polynomiale défocalisante. La preuve repose sur une inégalité de Strichartz généralisée sur des variétés munies d'une métrique de Lipschitz.

We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.

[1] S. Alinhac & P. Gérard - Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels, InterÉditions, 1991. | Zbl

[2] H. Bahouri & J.-Y. Chemin - « Équations d'ondes quasilinéaires et estimations de Strichartz », Amer. J. Math. 121 (1999), p. 1337-1377. | MR | Zbl

[3] V. Banica - « Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients », SIAM J. Math. Anal. 35 (2003), p. 868-883. | MR | Zbl

[4] J. Bourgain - Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, 1999. | MR | Zbl

[5] H. Brézis & T. Gallouet - « Nonlinear Schrödinger evolution equations », Nonlinear Anal. 4 (1980), p. 677-681. | MR | Zbl

[6] N. Burq, P. Gérard & N. Tzvetkov - « Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds », Amer. J. Math. 126 (2004), p. 569-605. | MR | Zbl

[7] N. Burq & F. Planchon - « Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications », J. Funct. Anal. 236 (2006), p. 265-298. | MR

[8] C. Castro & E. Zuazua - « Concentration and lack of observability of waves in highly heterogeneous media », Arch. Ration. Mech. Anal. 164 (2002), p. 39-72. | MR | Zbl

[9] T. Cazenave - Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University Courant Institute of Mathematical Sciences, 2003. | Zbl

[10] T. Cazenave & F. B. Weissler - « The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$ », Nonlinear Anal. 14 (1990), p. 807-836. | MR | Zbl

[11] D. Gilbarg & N. S. Trudinger - Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, Reprint of the 1998 edition. | MR | Zbl

[12] J. Ginibre & G. Velo - « The global Cauchy problem for the nonlinear Schrödinger equation revisited », Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), p. 309-327. | EuDML | Numdam | MR | Zbl

[13] -, « Scattering theory in the energy space for a class of nonlinear Schrödinger equations », J. Math. Pures Appl. (9) 64 (1985), p. 363-401. | MR | Zbl

[14] L. Hörmander - The analysis of linear partial differential operators. I, Classics in Mathematics, Springer, 2003. | MR | Zbl

[15] T. Kato - « On nonlinear Schrödinger equations », Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), p. 113-129. | EuDML | Numdam | MR | Zbl

[16] M. Keel & T. Tao - « Endpoint Strichartz estimates », Amer. J. Math. 120 (1998), p. 955-980. | MR | Zbl

[17] T. Ogawa & T. Ozawa - « Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem », J. Math. Anal. Appl. 155 (1991), p. 531-540. | MR | Zbl

[18] M. Reed & B. Simon - Methods of modern mathematical physics. I, second éd., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1980, Functional analysis. | MR | Zbl

[19] D. Salort - « Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients », Int. Math. Res. Not. (2005), p. 687-700. | MR | Zbl

[20] H. Smith & C. Sogge - « $L^q$ bounds for spectral clusters », in Phase Space Analysis of PDEs, Pisa, 2004. | MR | Zbl

[21] M. Spivak - A comprehensive introduction to differential geometry. Vol. One, Published by M. Spivak, Brandeis Univ., Waltham, Mass., 1970. | MR | Zbl

[22] G. Staffilani & D. Tataru - « Strichartz estimates for a Schrödinger operator with nonsmooth coefficients », Comm. Partial Differential Equations 27 (2002), p. 1337-1372. | MR | Zbl

[23] R. S. Strichartz - « Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations », Duke Math. J. 44 (1977), p. 705-714. | MR | Zbl

[24] C. Sulem & P.-L. Sulem - The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer, 1999, Self-focusing and wave collapse. | MR | Zbl

[25] D. Tataru - « Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation », Amer. J. Math. 122 (2000), p. 349-376. | MR | Zbl

[26] M. Tsutsumi - « On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions », Nonlinear Anal. 13 (1989), p. 1051-1056. | MR | Zbl

[27] M. Vladimirov - « On the solvability of mixed problem for a nonlinear equation of Schrödinger type », Sov. Math. Dokl. 29 (1984), p. 281-284. | Zbl

[28] K. Yajima - « Existence of solutions for Schrödinger evolution equations », Comm. Math. Phys. 110 (1987), p. 415-426. | MR | Zbl