Cauchy and Goursat problems for the generalized spin zero rest-mass fields on Minkowski spacetime

TL;DR

This paper investigates the well-posedness and decay properties of spin-zero massless fields on Minkowski spacetime using conformal geometric methods, establishing results for both Cauchy and Goursat problems.

Contribution

It introduces a conformal geometric approach to prove well-posedness and decay for spin-$n/2$ fields, extending existing methods to the Goursat problem on Minkowski spacetime.

Findings

Proves well-posedness of the Cauchy problem in Einstein's cylinder.

Establishes pointwise decay of the fields.

Demonstrates energy equalities between null boundaries and initial hypersurface.

Abstract

In this paper, we study the Cauchy and Goursat problems of the spin-$n/2$ zero rest-mass equations on Minkowski spacetime by using the conformal geometric method. In our strategy, we prove the wellposedness of the Cauchy problem in Einstein's cylinder. Then we establish pointwise decays of the fields and prove the energy equalities of the conformal fields between the null conformal boundaries $\scri^\pm$ and the hypersurface $\Sigma_0=\left\{ t=0 \right\}$. Finally, we prove the wellposedness of the Goursat problem in the partial conformal compactification by using the energy equalities and the generalisation of H\"ormander's result.