Local wellposedness for the 2+1-dimensional monopole equation

Czubak, Magdalena

doi:10.2140/apde.2010.3.151

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Local wellposedness for the 2+1-dimensional monopole equation

Magdalena Czubak

Vol. 3 (2010), No. 2, 151–174

DOI: 10.2140/apde.2010.3.151

Abstract

The space-time monopole equation on $ℝ^{2 + 1}$ can be derived by a dimensional reduction of the antiselfdual Yang–Mills equations on $ℝ^{2 + 2}$ . It can be also viewed as the hyperbolic analog of Bogomolny equations. We uncover null forms in the nonlinearities and employ optimal bilinear estimates in the framework of wave–Sobolev spaces. As a result, we show the equation is locally wellposed in the Coulomb gauge for initial data sufficiently small in $H^{s}$ for $s > \frac{1}{4}$ .

Keywords

monopole, null form, Coulomb gauge, wellposedness

Mathematical Subject Classification 2000

Primary: 35L70, 70S15

Milestones

Received: 10 February 2009

Revised: 15 September 2009

Accepted: 21 January 2010

Published: 8 June 2010

Authors

Magdalena Czubak
Department of Mathematics University of Toronto 40 Saint George Street Toronto, Ontario M5S 2E4 Canada
http://www.math.toronto.edu/czubak/

Vol. 3, No. 2, 2010