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/