Finite energy global well-posedness of the (3+1)-dimensional Yang-Mills equations using a novel Yang-Mill heat flow gauge

Abstract

In this thesis, we propose a novel choice of gauge for the Yang-Mills equations on the Minkowski space $\mathbb{R}^{1+d}$. A crucial ingredient is the associated Yang-Mills heat flow. Unlike the previous approaches, the new gauge is applicable for large data, while the special analytic structure of the Yang-Mills equations is still manifest.

As the first application of the new approach, we shall give new proofs of $H^{1}_{x}$ local well-posedness and finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. These are classical results first proved by S. Klainerman and M. Machedon using the method of local Coulomb gauges, which had been difficult to extend to other settings. As our approach does not possess its drawbacks (in particular the use of Uhlenbeck's lemma is avoided), it is expected to be more robust and easily applicable to other problems