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

Anne-Sophie de Suzzoni

doi:10.3934/cpaa.2014.13.991

Article Contents

2014, Volume 13, Issue 3: 991-1015. Doi: 10.3934/cpaa.2014.13.991

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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

Anne-Sophie de Suzzoni^1,

1.
University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise

Received: November 30, 2012

Revised: June 30, 2013

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: 35L05.

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