[Paper Review] Global solutions of two coupled Maxwell systems in the temporal gauge

Yuan Jian-jun|arXiv (Cornell University)|Mar 28, 2015

Advanced Mathematical Physics Problems5 references3 citations

TL;DR

This paper establishes the global existence of finite energy solutions for two coupled nonlinear Maxwell systems—Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs—in the temporal gauge. By leveraging gauge invariance and controlling the $\dot{H}^{1}$ norm of spatial gauge potentials via energy and $L^2$ norms, the authors prove that solutions exist globally in time for initial data in the finite energy space.

ABSTRACT

In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.

Motivation & Objective

• To establish the global existence of finite energy solutions for the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge.
• To extend previous results in the Coulomb gauge to the temporal gauge by exploiting gauge invariance and energy estimates.
• To control the $\dot{H}^1$ norm of spatial gauge potentials using energy and $L^2$ norms, ensuring global regularity.
• To address both topological and nontopological boundary conditions in the context of finite energy solutions.

Proposed method

• Utilizes gauge invariance to transform solutions from the Coulomb gauge to the temporal gauge.
• Applies energy conservation and $L^2$ bounds on the fields $A_i$, $\phi$, and $\widetilde{N}$ to control their growth.
• Employs the inequality $\|\nabla A\|_{L^2} \leq \|B\|_{L^2}$ for vector potentials with $B = \partial_1 A_2 - \partial_2 A_1$ to bound $\dot{H}^1$ norms.
• Uses Sobolev embedding and interpolation to bound $\|\nabla \phi\|_{L^2}$ in terms of $\|D_i \phi\|_{L^2}$, $\|\phi\|_{L^2}$, and $\|\nabla A^\mathrm{df}\|_{L^2}$.
• Establishes local well-posedness in $H^s \times H^{s-1}$ for $s \geq 1$ using space-time $X^{s,b}$ norms.
• Combines energy estimates with $L^2$ control to extend local solutions to global solutions in finite energy space.

Experimental results

Research questions

• RQ1 Can finite energy solutions of the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems be extended globally in time in the temporal gauge?
• RQ2 How can the $\dot{H}^1$ norm of spatial gauge potentials be controlled using only energy and $L^2$ norms in the temporal gauge?
• RQ3 What role does gauge invariance play in transferring global existence results from the Coulomb gauge to the temporal gauge?
• RQ4 How do nontopological and topological boundary conditions affect the global existence of solutions in the finite energy space?
• RQ5 What is the minimal regularity required to ensure global existence for these coupled Maxwell systems?

Key findings

• Global finite energy solutions exist for the Maxwell-Klein-Gordon system in the temporal gauge, extending prior results from the Coulomb gauge.
• Global solutions are established for the Maxwell-Chern-Simons-Higgs system in the temporal gauge under both nontopological and topological boundary conditions.
• The $\dot{H}^1$ norm of spatial gauge potentials is controlled by the energy and $L^2$ norms of the fields, enabling global existence via a priori estimates.
• Local well-posedness in $H^s \times H^{s-1}$ for $s \geq 1$ is proven using $X^{s,b}$ space techniques, with solutions in $C([-T,T]; H^s)$ for $T$ depending on initial data.
• The $L^2$ norms of $A_i$, $\phi$, and $\widetilde{N}$ are uniformly bounded in time by a function of $E(0)$ and $T$, ensuring no blow-up.
• The proof relies on energy conservation and interpolation inequalities to bound $\|\nabla \phi\|_{L^2}$ in terms of $\|D_i \phi\|_{L^2}$, $\|\phi\|_{L^2}$, and $\|\nabla A^\mathrm{df}\|_{L^2}$, closing the bootstrap argument.