Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in R1+3 \R^{1+3}

We prove that the initial value problem for the Dirac equation $\left ( -i\gamma^\mu \partial_\mu + m \right) \psi = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)\right) \psi \quad \text{in } \ \R^{1+3}$ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$, and an application of the $U^p$ and $V^p$-function spaces.Comment: 34 pages: To appear in SIAM J. Math. Analysi

Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge

We demonstrate null structure in the Yang-Mills equations in Lorenz gauge. Such structure was found in Coulomb gauge by Klainerman and Machedon, who used it to prove global well-posedness for finite-energy data. Compared with Coulomb gauge, Lorenz gauge has the advantage---shared with the temporal gauge---that it can be imposed globally in space even for large solutions. Using the null structure and bilinear space-time estimates, we also prove local-in-time well-posedness of the equations in Lorenz gauge, for data with finite energy. The time of existence depends on the initial energy and on the $H^s \times H^{s-1}$-norm of the initial potential, for some $s < 1$.Comment: Minor typos corrected, references update