The Cauchy problem for the coupled Klein–Gordon–Schrödinger system

Changxing Miao¹ ; Youbin Zhu¹

¹ Institute of Applied Physics and Computational Mathematics P.O. Box 8009, Beijing 100088, China

Annales Polonici Mathematici, Tome 89 (2006) no. 2, pp. 163-195

Résumé

We consider the Cauchy problem for a generalized Klein–Gordon–Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein–Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural restriction on the power of interactions. In the last part of this paper, we use special admissible pairs and Strichartz estimates to remove the restriction, thereby generalizing previous results and obtaining the well-posedness of the system.

Détail

DOI : 10.4064/ap89-2-6

Keywords: consider cauchy problem generalized klein gordon schr dinger system yukawa coupling prove existence global weak solutions compactness method through special choice admissible pairs match types equations prove uniqueness those solutions approach similar method presented ginibre velo pure klein gordon equation pure schr dinger equation though simple form method has unnatural restriction power interactions part paper special admissible pairs strichartz estimates remove restriction thereby generalizing previous results obtaining well posedness system