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The cubic nonlinear Dirac equation
Federico Cacciafesta1
1 SAPIENZA — Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy
Journées équations aux dérivées partielles (2012), article no. 1, 10 p.
  • Abstract

We present some results obtained in collaboration with prof. Piero D’Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in H 1 with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.

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DOI: 10.5802/jedp.84
Author's affiliations:
Federico Cacciafesta 1

1 SAPIENZA — Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy
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     author = {Federico Cacciafesta},
     title = {The cubic nonlinear {Dirac} equation},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {1},
     pages = {1--10},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2012},
     doi = {10.5802/jedp.84},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.84/}
}
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Federico Cacciafesta. The cubic nonlinear Dirac equation. Journées équations aux dérivées partielles (2012), article  no. 1, 10 p. doi : 10.5802/jedp.84. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.84/
  • References
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[1] Federico Cacciafesta. Global small solutions to the critical radial Dirac equation with potential. Nonlinear Analysis, 74 (2011), pp. 6060-6073. | MR | Zbl

[2] Federico Cacciafesta and Piero D’Ancona. Endpoint estimates and global existence for the nonlinear Dirac equation with a potential. http://arxiv.org/abs/1103.4014. | Zbl

[3] João-Paulo Dias and Mário Figueira. Global existence of solutions with small initial data in H s for the massive nonlinear Dirac equations in three space dimensions. Boll. Un. Mat. Ital. B (7), 1(3):861–874, 1987. | MR | Zbl

[4] Miguel Escobedo and Luis Vega. A semilinear Dirac equation in H s (R 3 ) for s>1. SIAM J. Math. Anal., 28(2):338–362, 1997. | MR | Zbl

[5] Daoyuan Fang and Chengbo Wang. Some remarks on Strichartz estimates for homogeneous wave equation. Nonlinear Anal., 65(3):697–706, 2006. | MR | Zbl

[6] Daoyuan Fang and Chengbo Wang. Weighted Strichartz estimates with angular regularity and their applications. 2008. | Zbl

[7] Chengbo Wang Jin-Cheng Jiang and Xin Yu. Generalized and weighted strichartz estimates. 2010. | Zbl

[8] Sergiu Klainerman and Matei Machedon. Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math., 46(9):1221–1268, 1993. | MR | Zbl

[9] Shuji Machihara, Makoto Nakamura, Kenji Nakanishi, and Tohru Ozawa. Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal., 219(1):1–20, 2005. | MR | Zbl

[10] Shuji Machihara, Makoto Nakamura, and Tohru Ozawa. Small global solutions for nonlinear Dirac equations. Differential Integral Equations, 17(5-6):623–636, 2004. | MR | Zbl

[11] Yves Moreau. Existence de solutions avec petite donnée initiale dans H 2 pour une équation de Dirac non linéaire. Portugal. Math., 46(suppl.):553–565, 1989. Workshop on Hyperbolic Systems and Mathematical Physics (Lisbon, 1988). | MR | Zbl

[12] Branko Najman. The nonrelativistic limit of the nonlinear Dirac equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 9(1):3–12, 1992. | Numdam | MR | Zbl

[13] Michael Reed. Abstract non-linear wave equations. Lecture Notes in Mathematics, Vol. 507. Springer-Verlag, Berlin, 1976. | MR | Zbl

[14] Jacob Sterbenz Angular regularity and Strichartz estimates for the wave equation. Int. Math. Res. Not. 2005, no. 4, 187Ð231. | MR | Zbl

[15] Bernd Thaller. The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. | MR | Zbl

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