The Global Existence and Uniqueness of Maxwell-Chern-Simons-Higgs Equation in (2+1) Dimensions
Mulyanto, Ardian N. Atmaja, Fiki T. Akbar, Bobby E. Gunara
TL;DR
The paper proves global existence and uniqueness of classical solutions for the Maxwell-Chern-Simons-Higgs system with a neutral scalar and a polynomial potential in (2+1) dimensions under the Coulomb gauge. By introducing a functional energy $\mathcal{J}(t)$ and employing energy estimates, Sobolev inequalities, and elliptic regularity, the authors control nonlinearities and propagate $H^{s}$ regularity for $s>0$ from finite-energy initial data. They derive detailed a priori bounds for $A_0$, $A$, $\phi$, and $N$, including higher-order derivatives up to fourth order, and extend local solutions to global ones via a continuation argument and density methods. The results enrich the understanding of planar gauge theories with Chern-Simons terms, providing rigorous global well-posedness and stability results that accommodate gauge interactions and nonlinear scalar potentials.
Abstract
In this paper, we show the global existence and uniqueness of classical solutions of the Maxwell-Chern-Simmons-Higgs system coupled to a neutral scalar with nontrivial scalar potential on (2+1) dimensional Minkowski spacetime. Our methods rely only on classical existence theorems, including energy estimates, the Sobolev inequality, and the choice of the Coulomb gauge condition. The equations are well-posed for finite initial data and the solution preserves any additional $H^{s}$ regularity for $s>0$ in the data.