Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation

Revista Matematica Iberoamericana

Volumn 19, Issue 1, 2003, Pages 179-194

  Machihara, Shuji ^a   Nakanishi, Kenji ^b   Ozawa, Tohru ^c  

^a SHIMANE UNIVERSITY   (Japan)

^b NAGOYA UNIVERSITY   (Japan)

^c HOKKAIDO UNIVERSITY   (Japan)

Author keywords

Nonlinear Dirac equation; Nonlinear Schr dinger equation; Nonrelativistic limit; Strichartz's estimate

Indexed keywords

EID: 0037570590     PISSN: 02132230     EISSN: 22350616     Source Type: Journal    
DOI: 10.4171/RMI/342     Document Type: Article

References (25)

1. BERGH, J. AND LÖFSTRÖM, J.: Interpolation spaces. Springer, Berlin-Heiderberg-New York, 1976.
2. BRENNER, P.: On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations. J. Differential Equations 56 (1985), 310-344.
3. s. Nonlinear Anal. 14 (1990), 807-836.
4. s for the massive nonlinear Dirac equations in three space dimensions. Boll. Un. Mat. Ital. 3(7) 1 (1987), 861-874.
5. 3) for s > 1. SIAM J. Math. Anal. 2 (1997), 338-362.
6. FINKELSTEIN, R., LELEVIER, R. AND RUDERMAN, M.: Nonlinear spinor fields. Phys. Rev. 83 (1951), 326-332.
7. GINIBRE, J. AND VELO, G.: Time decay of finite energy solutions of the non linear Klein-Gordon and Schrödinger equations. Ann. List. H. Poincaré Phys. Théor. 43 (1985), 399-442.
8. GINIBRE, J. AND VELO, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144 (1992), 163-188.
9. GINIBRE, J. AND VELO, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995), 50-68.
10. KEEL, M. AND TAO, T.: Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955-980.
11. LINDBLAD, H.: A sharp counterexample to local existence of low regularity solutions to nonlinear wave equations. Duke Math. J. 72 (1993), 503-539.
12. LINDBLAD, H.: Counterexamples to local existence for semi-linear wave equations. Amer. J. Math. 118 (1996), 1-16.
13. LINDBLAD, H. AND SOGGE, C. D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), 357-426.
14. MACHIHARA, S., NAKANISHI, K. AND OZAWA, T.: Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann. 322 (2002), 603-621.
15. MATSUYAMA, T.: A remark on the nonrelativistic limit for semilinear Dirac equations. Nonlinear Anal. 25 (1995), 1139-1146.
16. MATSUYAMA, T.: Rapidly decreasing solutions and nonrelativistic limit of semilinear Dirac equations. Rev. Math. Phys. 7 (1995), 243-267.
17. NAJMAN, B.: The nonrelativistic limit of the nonlinear Klein-Gordon equation. Nonlinear Anal. 15 (1990), 217-228.
18. NAJMAN, B.: The nonrelativistic limit of the Klein-Gordon and Dirac equations. In Differential Equations with Applications in Biology, Physics and Engineering, J. Goldstein, F. Kappel, W. Schappacher (Eds.), Lect. Notes Pure Appl. Math. 133 (1991), 291-299, Marcel Dekker.
19. NAJMAN, B.: The nonrelativistic limit of the nonlinear Dirac equation. Ann. Inst. H. Poincaré, Anal. Non linéaire 9 (1992), 3-12.
20. PECHER, H.: Nonlinear small data scattering for the wave and Klein-Gordon equation. Math. Z. 185 (1984), 261-270.
21. PONCE, G. AND SIDERIS, T. C.: Local regularity of nonlinear wave equations in three space dimensions. Comm. Partial Differential Equations 18 (1993), 169-177.
22. RAÑADA, A. F.: Classical nonlinear Dirac field models of extended particles. In Quantum Theory, Groups, Fields and Particles. D. Reidel, Amsterdam, 1982.
23. SOLER, M.: Classical, stable nonlinear spinor field with positive rest energy. Phys. Rev. D l (1970), 2766-2769.
24. WAKANO, M.: Intensely localized solutions of the classical Dirac-Maxwell field equations. Progr. Theoret. Phys. 35 (1966), 1117-1141.
25. s. Nonlinear Anal. 31 (1998), 573-587.