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

Abstract

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 wellposedness for finite-energy data in the temporal gauge by passing to local Coulomb gauges via Uhlenbeck’s Lemma. Compared with the Coulomb gauge, the 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 wellposedness of the Yang–Mills equations in Lorenz gauge for data with finite energy, with a time of existence depending on the initial energy and on the $H^s \times H^{s–1}$-norm of the initial gauge potential, for some choice of $s < 1$ sufficiently close to 1.