May  2014, 13(3): 991-1015. doi: 10.3934/cpaa.2014.13.991

Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space

1. 

University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Received  December 2012 Revised  July 2013 Published  December 2013

In this paper, the almost sure global well-posedness of the cubic non linear wave equation on the sphere is studied when the initial datum is a random variable with values in low regularity spaces. The result is first proved on the 3D sphere, thanks to the existence of a uniformly bounded in $L^p$ basis of $L^2(S^3)$ and then it is extended to $R^3$ thanks to the Penrose transform.
Citation: Anne-Sophie de Suzzoni. Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space. Communications on Pure & Applied Analysis, 2014, 13 (3) : 991-1015. doi: 10.3934/cpaa.2014.13.991
References:
[1]

N. Burq and G. Lebeau, Injections de Sobolev Probabilistes et Applications,, preprint., ().   Google Scholar

[2]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation,, preprint., ().   Google Scholar

[3]

Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, \emph{Invent. Math.}, 173 (2008), 449.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[4]

D. Christodoulou, Global solutions of non linear hyperbolic equations for small initial data,, \emph{Comm. Pure. Appl. Math.}, 39 (1986), 267.  doi: 10.1002/cpa.3160390205.  Google Scholar

[5]

A-S. de Suzzoni, Large data low regularity scattering results for the wave equation on the Euclidian space,, \emph{Comm. PDE}, 38 (2013), 1.  doi: 10.1080/03605302.2012.736910.  Google Scholar

[6]

X. Fernique, R\'egularit\'e des trajectoires des fonctions al\'eatoires gaussiennes,, \emph{Ecole d’\'et\'e St. Flour.} {IV}-1974, 480 (1975), 1.   Google Scholar

[7]

Michel Ledoux, The Concentration of Measure Phenomenon,, Mathematical Surveys and Monographs, (2001).   Google Scholar

[8]

N. Tzvetkov, Remark on the Null-condition for the nonlinear wave equation,, \emph{Bollettino U.M.I.}, 8 (2000), 135.   Google Scholar

show all references

References:
[1]

N. Burq and G. Lebeau, Injections de Sobolev Probabilistes et Applications,, preprint., ().   Google Scholar

[2]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation,, preprint., ().   Google Scholar

[3]

Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, \emph{Invent. Math.}, 173 (2008), 449.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[4]

D. Christodoulou, Global solutions of non linear hyperbolic equations for small initial data,, \emph{Comm. Pure. Appl. Math.}, 39 (1986), 267.  doi: 10.1002/cpa.3160390205.  Google Scholar

[5]

A-S. de Suzzoni, Large data low regularity scattering results for the wave equation on the Euclidian space,, \emph{Comm. PDE}, 38 (2013), 1.  doi: 10.1080/03605302.2012.736910.  Google Scholar

[6]

X. Fernique, R\'egularit\'e des trajectoires des fonctions al\'eatoires gaussiennes,, \emph{Ecole d’\'et\'e St. Flour.} {IV}-1974, 480 (1975), 1.   Google Scholar

[7]

Michel Ledoux, The Concentration of Measure Phenomenon,, Mathematical Surveys and Monographs, (2001).   Google Scholar

[8]

N. Tzvetkov, Remark on the Null-condition for the nonlinear wave equation,, \emph{Bollettino U.M.I.}, 8 (2000), 135.   Google Scholar

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