GLOBAL EXISTENCE for CRITICAL POWER YANG-MILLS-HIGGS EQUATIONS
by
Mark Keel, UCLA
JWB 335, 4:15pm Friday, February 16, 1996
Abstract
The talk will present two global existence and uniqueness theorems
for the critical power Yang-Mills-Higgs equations in RxR^3.
These theorems extend the known results by allowing a critical power
Higgs nonlinearity for both smooth and finite energy initial data.
In particular, we allow Higgs nonlinearities in the Lagrangian which
behave like $V(\phi) = \norm{\phi}^6$ for large $\norm{\phi}$. This
corresponds to a degree 5 power nonlinearity in the equations
themselves.
Some of the ideas for the results are easily seen in the context of
the wave equation. I'll first show a simple extension of known
non-concentration results for the critical power wave equation
(CPWE)
\square \phi = \norm{\phi}^4 \phi
for (t,x) in RxR^3 which will be necessary for the Yang - Mills -
Higgs results. I'll then give a simple proof of global existence for
the CPWE.
As for the Yang-Mills-Higgs results: I first indicate why the Higgs
field enjoys the same local non-concentration as the CPWE. Finally,
I'll describe how to take advantage of this by using the local
estimates of Klainerman-Machedon [Comm. Pure and Applied Math, 1993,
1221-1268] and their application to the Yang-Mills Equations [K-M,
Annals of Math, 1995, 39-119].
Requests for preprints and reprints to: rmm@math.utah.edu Attn: Mark Keel
This source can be found at http://www.math.utah.edu/research/