Abstract
We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H s(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H s(M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.
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Ayache, A., Tzvetkov, N.: L p properties of Gaussian random series. Trans. Am. Math. Soc. www.ams.org/lion/0000-000-00/S0002-9947-08-04456-5/home.htms (to appear)
Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166, 1–26 (1994)
Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Commun. Math. Phys. 176, 421–445 (1996)
Burq, N., Gérard, P., Tzvetkov, N.: An instability property of the nonlinear Schrödinger equation on S d. Math. Res. Lett. 9(2–3), 323–335 (2002)
Burq, N., Gérard, P., Tzvetkov, N.: Two singular dynamics of the nonlinear Schrödinger equation on a plane domain. Geom. Funct. Anal. 13(1), 1–19 (2003)
Burq, N., Gérard, P., Tzvetkov, N.: Multilinear eigenfunctions estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Sci. Éc. Norm. Supér., IV. Sér. 38, 255–301 (2005)
Burq, N., Lebeau, G., Planchon, F.: Global existence for energy critical waves in 3-d domains. J. Am. Math. Soc. http://fr.arxiv.org/abs/math/0607631 (2008, to appear)
Burq, N., Tzvetkov, N.: Invariant measures for a three dimensional nonlinear wave equation. Int. Math. Res. Not. 2007, rnm108-26 (electronic)
Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations II: A global result. Invent. Math. http://arxiv.org/abs/0707.1448 (2008, to appear)
Christ, M., Colliander, J., Tao, T.: Ill-posedness for nonlinear Schrödinger and wave equations. (2003, preprint)
Kapitanskii, L.: Some generalizations of the Strichartz–Brenner inequality. Leningrad Math. J. 1, 693–726 (1990)
Lebeau, G.: Perte de régularité pour les equations d’ondes sur-critiques. Bull. Soc. Math. Fr. 133, 145–157 (2005)
Paley, E.A.C.R., Zygmund, A.: On some series of functions (1) (2) (3). Proc. Camb. Philos. Soc. 26, 337–357, 458–474, (1930); 28, 190–205 (1932)
Smith, H., Sogge, C.: On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math. 198(1), 107–153 (2006)
Sogge, C.: Concerning the L p norm of spectral clusters for second order elliptic operators on compact manifolds. J. Funct. Anal. 77, 123–138 (1988)
Tzvetkov, N.: Invariant measures for the nonlinear Schrödinger equation on the disc. Dyn. Partial Differ. Equ. 3, 111–160 (2006)
Tzvetkov, N.: Invariant measures for the defocusing NLS. Ann. Inst. Fourier (2008, to appear)
Tzvetkov, N.: Construction of a Gibbs measure associated to the periodic Benjamin–Ono equation. (2006, preprint)
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Mathematics Subject Classification (2000)
35Q55, 35BXX, 37K05, 37L50, 81Q20
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Burq, N., Tzvetkov, N. Random data Cauchy theory for supercritical wave equations I: local theory. Invent. math. 173, 449–475 (2008). https://doi.org/10.1007/s00222-008-0124-z
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DOI: https://doi.org/10.1007/s00222-008-0124-z