Global well-posedness and scattering for the massive Dirac-Klein-Gordon system in two dimensions

TL;DR

The paper proves global well-posedness and scattering for the two-dimensional massive Dirac-Klein-Gordon system with small, low-regularity data under a mass non-resonance condition $2M>m>0$. It introduces a global-in-time iteration built on a refined resolution space that fuses localized Strichartz estimates with generalized $X^{s,b}$-type norms, including a novel high-modulation structure, and leverages a modulation-null structure analysis to control nonlinear interactions. The main result establishes global solutions with data in $\psi_0\in H^{\frac12+\varepsilon}$ and $(\phi_0,\phi_1)\in H^{1+\varepsilon}\times H^{\varepsilon}$ for any $\varepsilon>0$, along with linear scattering to free solutions as $t\to\pm\infty$, without requiring spatial decay of the data. This work extends the 2D theory beyond energy conservation-based methods by providing a robust, fully nonlocal framework that handles derivative-type nonlinearities and multi-speed dynamics, potentially adaptable to other coupled relativistic systems.

Abstract

We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small and low regularity initial data in dimension two. To achieve this, we impose a non-resonance condition on the masses.

Theorems & Definitions (18)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof