On the radius of spatial analyticity for solutions of the Dirac–Klein–Gordon equations in two space dimensions

Abstract

We consider the initial value problem for the Dirac–Klein–Gordon equations in two space dimensions. Global regularity for $C^{\infty }$ data was proved by Grünrock and Pecher. Here we consider analytic data, proving that if the initial radius of analyticity is ${\sigma }_0>0$, then for later times $t>0$ the radius of analyticity obeys a lower bound $\sigma (t)\ge {\sigma }_0\exp (-At)$. This provides information about the possible dynamics of the complex singularities of the holomorphic extension of the solution at time $t$. The proof relies on an analytic version of Bourgain's Fourier restriction norm method, multilinear space–time estimates of null form type and an approximate conservation of charge.