## Low regularity local solutions for field equations.(English)Zbl 0852.35098

Summary: We prove local existence and uniqueness of low regularity solutions to semilinear systems of coupled wave equations and of coupled Dirac-Klein-Gordon equations. The smoothness of our solutions is below the classical level. The main lemma is an $$L^2$$ estimate for products of solutions of linear equations inspired by earlier work by Strichartz, Klainerman and Machedon.

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35L55 Higher-order hyperbolic systems

### Keywords:

coupled wave equations; Dirac-Klein-Gordon equations
Full Text:

### References:

 [1] Bachelot A., Ann. IHP Anal. non lin. 1 pp 453– (1984) [2] Bachelot A., Ann. IHP Phys. Theor 48 pp 383– (1988) [3] Beals M., Memoirs AMS 264 38 (1982) [4] Beals M., Journées EDP St. Jean de Monts, (1992) [5] Bézard M., Note C.R. Acad. Sc. Paris 314 pp 1241– (1992) [6] Björken J., Relativistic Quantum Mechanics (1964) [7] DOI: 10.1007/BF01215290 · Zbl 0321.35052 [8] DOI: 10.1007/BF01258907 · Zbl 0457.35059 [9] Ginibre J., Ann. IHP Phys. Theor. 36 pp 59– (1982) [10] Greiner C., Relativistic Quantum Mechanics (1990) · Zbl 0718.35078 [11] Hörmander L., The Analysis of Linear Partial Differential Operators (1985) [12] Hörmander L., Lecture notes (1988) [13] DOI: 10.1007/BF00251584 · Zbl 0361.35046 [14] DOI: 10.1215/S0012-7094-87-05518-9 · Zbl 0644.35012 [15] Klainenman S., AMS Lect. Appl.Math. 23 pp 293– (1986) [16] DOI: 10.1080/03605309908820708 · Zbl 0712.35018 [17] Marshall B., J.Math. Pures Appl. 59 pp 417– (1980) [18] DOI: 10.1016/0022-1236(80)90110-X · Zbl 0442.35017 [19] DOI: 10.1080/03605309308820925 · Zbl 0803.35096 [20] DOI: 10.2307/1970347 · Zbl 0204.16004 [21] DOI: 10.2307/2946554 · Zbl 0836.35096 [22] DOI: 10.1007/BF01245080 · Zbl 0754.35004 [23] DOI: 10.1215/S0012-7094-77-04430-1 · Zbl 0372.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.