Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

2009/10/09 by Keel, Markus, Roy, Tristan, Tao, Terence
#35L10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.1850

Abstract

We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in H^s_x in the Coulomb gauge for all s > √(3)/2 ≈ 0.866. This extends previous work of Klainerman-Machedon \citekl-mac:mkg on finite energy data s ≥ 1, and Eardley-Moncrief \citeeardley for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of H^s_x rather than H¹_x. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the L²_x norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.

Related

• Unconditional well-posedness below energy norm for the Maxwell-Klein-Gordon system
• Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon\n equation at energy regularity
• Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in 1+4 dimensions
• Global well-posedness for the massive Maxwell-Klein-Gordon equation with\n small critical Sobolev data
• Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge