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Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions. (English) Zbl 1129.35066

The authors start with considering the coupled Dirac-Klein-Gordon system of equations (DKG) on ${ℝ}^{n+1}$, and, in the particular case $n=2$, they prove the local well-posedness of such a system for the Cauchy problem with initial data belonging to a suitable range of Sobolev spaces. After an introduction emphasizing the main result (Theorem 1.1), the paper splits into seven sections in order to prove Theorem 1.1 and some related results.

Reviewer’s remark. Let us recall that local well-posedness means that there exists a unique local solution that depends continuously on the initial Cauchy conditions. With this respect, it may be useful to underline that the local uniqueness of solutions is usually easily obtained for Cauchy problems in nonlinear (or quasilinear) PDE’s, when solutions are restricted to belong to suitable Sobolev spaces. (See the rich literature on this subject.) It should be more worthwhile to investigate the formal geometric properties of the system (DKG). In fact such a system is analytic. Therefore we can relate smooth solutions to singular and weak ones. Furthermore, by considering the integral bordism groups of (DKG) we can obtain existence of global solutions. Today, however, the main interest in the (DKG) equation is not in the commutative case, but in a noncommutative framework, since (DKG) has been introduced to describe subatomic interactions ...

MSC:
 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to quantum-mechanical equations 35L70 Nonlinear second-order hyperbolic equations