The Yang-Mills-Higgs heat flow on \(\mathbb{R}^ 3\) (Q1207157)

The purpose of this paper is to study the `heat flow' of the Lagrangian, that is, the evolution equation \(\partial_ t(A,\varphi)=- \text{grad}({\mathcal L})\) and show reasonable conditions under which it converges to a monopole. The conditions arrived at by investigations in steps, are: i) short time existence of the flow, (ii) evolution equations for various quantities, (iii) long time existence of the flow, (iv) an analysis of the decay solutions of the model problem \(\partial_ tu=\Delta u\), and finally (v) proof of the power law decay of the Bogomolny tensor along with all its derivatives and convergence of \((A,\varphi)\). The main technical tools used are the parabolic maximum principle, and Sobolev and interpolation inequalities.