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

TL;DR

This paper investigates how choosing a specific basis of the $L^2(S^3)$ space affects the almost sure global well-posedness of the cubic nonlinear wave equation on the sphere and Euclidean space, using probabilistic methods.

Contribution

It introduces a basis with uniform $L^p$ bounds for $L^2(S^3)$ and extends well-posedness results from the sphere to Euclidean space via the Penrose transform.

Findings

Almost sure global well-posedness established for the cubic wave equation on $S^3$.

Extension of results to $ ^3$ using the Penrose transform.

Identification of a basis of $L^2(S^3)$ with uniform $L^p$ bounds.

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 domain is first the 3D sphere, thanks to the existence of a uniformly bounded in $L^p$ basis of $L^2(S^3)$ and then the result is extended to $\R^3$ thanks to the Penrose transform.