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

Abstract

Abstract. In recent work, the authors extended the local and global well-posedness theory for the 1D Dirac-Klein-Gordon equations, but the uniqueness of the solutions was only known in the contraction spaces (of Bourgain-Klainerman-Machedon type). Here we prove some unconditional uniqueness results [that is, uniqueness in the larger space C([0, T ]; X0), where X0 denotes the data space]. We also prove a result about persistence of higher regularity, which is stronger than the standard version obtained from the contraction argument, since our result allows to independently increase the regularity of the spinor and scalar fields, whereas in the standard result they must be increased by the same amount.

On the radius of spatial analyticity for the 1d Dirac–Klein–Gordon equations

“…Proof. By Duhamel's formula, as in the proof of Theorem 2.2 in [14], one can reduce to the case F = G = 0, and this case is easily proved by changing to characteristic coordinates or by using Plancherel's theorem as in [21,Lemma 2]. Replacing u by its complex conjugate u does not affect the argument, since u(t) = e −itDx f implies u = e −itDx f , as one can check on the Fourier transform side.…”

“…Now fix such a T and let σ ∈ (0, σ 0 ] be a parameter to be chosen; then (21) of course holds also with σ 0 replaced by σ. Let A ≫ 1 denote a constant which may depend on M , M σ0 (0) and N σ0 (0); the choice of A will be made explicit below.…”

“…x ) −s/2 L 2 (R), s ∈ R, has been intensively studied; see [8,4,2,5,18,15,21,20,19,23,16,22,6]. Local well-posedness holds for data (3) (ψ 0 , φ 0 , φ 1 ) ∈ H s (R; C 2 ) × H r (R; R) × H r−1 (R; R)…”

Unconditional uniqueness of solution forH˙sccritical NLS in high dimensions

“…For other nonlinear evolution equations, we can refer (UU) results to [1,11,13,6,16] in the case of wave equation, to [5] in the case of Navier-Stokes system, to [2] in the case of Benjamin-Ono equation, to [10,9] in the case of Zakharov system and Maxwell-Dirac equation, to [12] in the case of Klein-Gorden-Schödinger system and to [15] in the case of Dirac-Klein-Gorden equations.…”

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].…”