Local wellposedness for the 2+1 dimensional monopole equation

Details Local wellposedness for the 2+1 dimensional monopole equation Local wellposedness for the 2+1 dimensional monopole equation

2007-12-10

arxiv.org/abs/0712.1393v2

Mathematics

Analysis of PDEs

23 pages; Added some remarks, and rewrote parts of Sections 4 and 5; Submitted

Scientific paper

The space-time monopole equation on $\R^{2+1}$ can be derived by a dimensional reduction of the anti-self-dual 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 $s>{1/4}$.

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