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

## TL;DR

This paper studies how the radius of spatial analyticity of solutions to the Dirac-Klein-Gordon equations in two dimensions evolves over time, establishing an exponential lower bound based on initial analyticity.

## Contribution

It proves a lower bound on the radius of analyticity for solutions with analytic initial data, extending understanding of complex singularity dynamics in these equations.

## Key findings

- Radius of analyticity decays at most exponentially over time
- Provides a quantitative lower bound on the analyticity radius
- Uses an analytic adaptation of Bourgain's method and multilinear estimates

## 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 Gr\"unrock 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.