Global Well-Posedness of the 1d Dirac–klein–gordon System in Sobolev Spaces of Negative Index

Abstract

Abstract. We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of "almost conservation law" and "I-method" introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.

Remarks on regularity and uniqueness of the Dirac–Klein–Gordon equations in one space dimension

“…The hypotheses of the lemma are satisfied since r > 1/6, r ≥ s and r > 1/6 − 2s/3 (by the definition of R). From (3.9) (as well as its counterpart for φu) and (1.4) we finally conclude that 10) and together with (3.8) this proves (3.2) in the case r ≤ 1/2 (recall that σ = ρ in this case).…”

“…In fact, for s ≥ 0, any local wellposedness result can be extended to a global one by using the conservation of charge; see [1,2,[5][6][7]. Global well-posedness for a range of negative s was obtained first in [8], and then for a larger range of r (but the same range of s) in [10].…”

“…Since the work of Bournaveas [1], there has been a lot of progress Vol. 17 (2010) Regularity and uniqueness for 1D DKG 455 on lowering the initial data regularity required for global or local well-posedness; see [2,[5][6][7][8][9][10]. Our aim here is to improve the statements of uniqueness and persistence of higher regularity in the existing local well-posedness theory, which we recall in the following theorem: …”

Sharp ill-posedness of the Dirac–Klein–Gordon system in one dimension

“…Other earlier papers which obtained the well-posedness in subsets of the region |s| ≤ r ≤ s + 1 are [2,3,4,6,7,8,9,10,15,22,23,24]. There are also global wellposedness results with s < 0 in which they used Bourgain's frequency decomposition technique or I-method with a help of the charge conservation law [5,25,27]. Previous attempts at the critcal point.…”

Global well-posedness of the Maxwell–Dirac system in two space dimensions

“…Since then a number of papers improving the local and global theory for 1d DKG have appeared, see [13,1,22,[26][27][28]31,32,23].…”