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

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.

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

“…Remark 6. The Goursat problems for wave equations on spinor fields were also established in some other works [53,54,64,65].…”