Abstract
This article is devoted to the study of a quasilinear Schrödinger equation coupled with an elliptic equation on the metric g. We first prove that, in this context, the propagation of regularity holds which ensures local wellposedness for initial data small enough in \(\dot H^{\tfrac{1} {2}} \) and belonging to the Besov space \(\dot B_{2,1}^{\tfrac{3} {2}} \). In a second step, we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence which only involves the \(\dot B_{2,\infty }^{1 + \varepsilon } \) norm on the initial data.
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References
Anker J-P, Pierfelice V. Nonlinear Schrödinger equation on real hyperbolic spaces. Ann l’Institut Henri Poincare Non Linear Anal, 2009, 26: 1853–1869
Anton R. Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bull Soc Math France, 2008, 136: 27–65
Bahouri H, Chemin J-Y. Équations d’ondes quasilinéaires et inégalités de Strichartz. Amer J Math, 1999, 121: 1337–1377
Bahouri H, Chemin J-Y. Équations d’ondes quasilinéaires et effet dispersif. Internat Math Res News, 1999, 21: 1141–1178
Bahouri H, Chemin J-Y. Microlocal analysis, bilinear estimates and cubic quasilinear wave equation. Astéristique, 2003, 248: 93–141
Bahouri H, Chemin J-Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren Math Wissenschaften, vol. 343. Berlin: Springer, 2011
Banica V. Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. SIAM J Math Anal, 2003, 35: 868–883
Bejenaru I, Ionescu A, Kenig C, et al. Global Schrödinger maps in dimensions d ⩾ 2: Small data in the critical Sobolev spaces. Ann Math, 2011, 173: 1443–1506
de Bouard A, Hayashi N, Saut J-C. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun Math Phys, 1997, 189: 73–105
Bouclet J-M, Tzvetkov N. Estimates for long range perturbations. Amer J Math, 2007, 129: 1565–1609
Burq N. Estimations de Strichartz pour des perturbations longue portée de l’opérateur de Schrödinger. Séminaire E.D.P de l’Ecole polytechnique, 2001–2002
Burq N, Gérard P, Tzvetkov N. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer J Math, 2004, 126: 569–605
Burq N, Planchon F. Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications. J Funct Anal, 2006, 236: 265–298
Coifman R, Meyer Y. Au delà des opérateurs pseudo-différentiels (in French). [Beyond Pseudodifferential Operators] With an English summary. Astérisque, 57. Paris: Soc Math France, 1978
Colin M. On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension. Comm Partial Differential Equations, 2002, 27: 325–354
Ginibre J, Velo G. Smoothing properties and retarded estimates for some dispersive evolution equations. Commun Math Phys, 144, pages 163–188.
Ichinose W. Some remarks on the Cauchy problem for Schrödinger type equations. Osaka J Math, 1984, 21: 565–581
Keel M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120: 995–980
Kenig C E, Ponce G, Vega L. The Cauchy problem for quasi-linear Schrödinger equations. Invent Math, 2004, 158: 343–388
Marzuola J, Metcalfe J, Tataru D. Quasilinear Schrödinger equations I: Small data and quadratic interactions. Adv Math, 2012, 231: 1151–1172
Marzuola J, Metcalfe J, Tataru D. Quasilinear Schrödinger equations II: Small data and cubic nonlinearities. Kyoto J Math, 2015, in press
Mizohata S. On some Schrödinger type equations. Japan Acad Proc Ser A Math Sci, 1981, 57: 81–84
Robbiano L, Zuily C. Strichartz estimates for Schrödinger equations with variable coefficients. Mémoires de la Société Mathématique de France, 2005, 101–102, 208pp
Salort D. Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients. Internat Math Res Not, 2005, 11: 687–700
Salort D. Dispersion and Strichartz estimates for the Liouville equation. J Differential Equations, 2007, 233: 543–584
Staffilani G, Tataru D. Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm Partial Differential Equations, 2002, 27: 1337–1372
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Chemin, JY., Salort, D. Wellposedness of some quasi-linear Schrödinger equations. Sci. China Math. 58, 891–914 (2015). https://doi.org/10.1007/s11425-015-4993-5
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DOI: https://doi.org/10.1007/s11425-015-4993-5