# zbMATH — the first resource for mathematics

On the energy critical Schrödinger equation in 3D non-trapping domains. (English) Zbl 1200.35066
Summary: We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for $$H_0^1 (\Omega)$$ data on any smooth, non-trapping domain $$\Omega \subset \mathbb R^3$$. The key ingredient is a smoothing effect in $$L_x^5(L_t^2)$$ for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35Q55 NLS equations (nonlinear Schrödinger equations) 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35P25 Scattering theory for PDEs
Full Text:
##### References:
 [1] Anton, Ramona, Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains, J. math. pures appl. (9), 89, 4, 335-354, (2008) · Zbl 1148.35081 [2] Burq, N.; Gérard, P.; Tzvetkov, N., On nonlinear Schrödinger equations in exterior domains, Ann. inst. H. Poincaré anal. non linéaire, 21, 3, 295-318, (2004) · Zbl 1061.35126 [3] Burq, Nicolas, Estimations de Strichartz pour des perturbations à longue portée de l’opérateur de Schrödinger, () · Zbl 1292.35188 [4] Burq, Nicolas; Lebeau, Gilles; Planchon, Fabrice, Global existence for energy critical waves in 3-D domains, J. amer. math. soc., 21, 3, 831-845, (2008) · Zbl 1204.35119 [5] Burq, Nicolas; Planchon, Fabrice, Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications, J. funct. anal., 236, 1, 265-298, (2006) · Zbl 1293.35264 [6] Burq, Nicolas; Planchon, Fabrice, Global existence for energy critical waves in 3-D domains: Neumann boundary conditions, Amer. J. math., 131, 6, 1715-1742, (2009) · Zbl 1184.35210 [7] Cazenave, Thierry; Weissler, Fred B., The Cauchy problem for the critical nonlinear Schrödinger equation in $$H^s$$, Nonlinear anal., 14, 10, 807-836, (1990) · Zbl 0706.35127 [8] Christ, Michael; Kiselev, Alexander, Maximal functions associated to filtrations, J. funct. anal., 179, 2, 409-425, (2001) · Zbl 0974.47025 [9] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $$\mathbb{R}^3$$, Ann. of math. (2), 167, 3, 767-865, (2008) · Zbl 1178.35345 [10] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. inst. H. Poincaré anal. non linéaire, 2, 4, 309-327, (1985) · Zbl 0586.35042 [11] Ivanovici, Oana, Precise smoothing effect in the exterior of balls, Asymptot. anal., 53, 4, 189-208, (2007) · Zbl 1387.35138 [12] Ivanovici, Oana, Counter example to Strichartz estimates for the wave equation in domains, Math. ann., 347, 627-673, (2010) · Zbl 1201.35060 [13] Ivanovici, Oana, On the schrodinger equation outside strictly convex obstacles, (2008), Analysis & PDE, in press · Zbl 1222.35186 [14] Ivanovici, Oana; Planchon, Fabrice, Square function and heat flow estimates on domains, (2008) · Zbl 1200.35066 [15] Planchon, Fabrice, Dispersive estimates and the 2D cubic NLS equation, J. anal. math., 86, 319-334, (2002) · Zbl 1034.35130 [16] Planchon, Fabrice; Vega, Luis, Bilinear virial identities and applications, Ann. sci. école. norm. sup., 42, 261-290, (2009) · Zbl 1192.35166 [17] Smith, Hart F.; Sogge, Christopher D., On the critical semilinear wave equation outside convex obstacles, J. amer. math. soc., 8, 4, 879-916, (1995) · Zbl 0860.35081 [18] Smith, Hart F.; Sogge, Christopher D., On the $$L^p$$ norm of spectral clusters for compact manifolds with boundary, Acta math., 198, 1, 107-153, (2007) · Zbl 1189.58017 [19] Staffilani, Gigliola; Tataru, Daniel, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. partial differential equations, 27, 7-8, 1337-1372, (2002) · Zbl 1010.35015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.