A spinor-adapted geometric approach for nonlinear Dirac systems and its application to a tensorial wave-Dirac system near Minkowski spacetime
We study a nonlinear tensorial wave-Dirac system on $(1+3)$-dimensional asymptotically flat spacetimes as a semilinear model motivated by the Maxwell-Dirac and Einstein-Dirac systems. The purpose of this model is to isolate the interaction between the null geometry of antisymmetric tensor fields and the intrinsic first-order geometry of the Dirac equation while avoiding the derivative loss mechanism of the Einstein equations and the gauge structure of the Maxwell equations. Our analysis preserves the first-order nature of the Dirac equation throughout the nonlinear argument. The Dirac current provides the fundamental energy identity, while quantitative spacetime estimates are obtained from the wave equation arising from the squared Dirac operator. Combining these ingredients with integrated local energy decay estimates and $r^p$-weighted energy hierarchies, we establish a coupled energy method for the tensorial and spinorial components. A key observation is that the Clifford algebra is compatible with the null decomposition of antisymmetric tensor fields and excludes the most singular nonlinear interactions. As a consequence, we establish the global existence of small-data solutions together with quantitative weighted energy and decay estimates. Remarkably, combining the null structure with dyadic argument, weak decay such as $t^{-\frac12-\delta}$ is sufficient to obtain nonlinear stability of the system, in the spirit of \cite{DHRT}. We expect that the geometric ideas developed in this paper provide a useful starting point for the study of more general nonlinear Dirac systems, including the Maxwell-Dirac and Einstein-Dirac equations.