The cubic nonlinear Dirac equation

Federico Cacciafesta[1]

  • [1] SAPIENZA — Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy

Journées Équations aux dérivées partielles (2012)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Cacciafesta, Federico. "The cubic nonlinear Dirac equation." Journées Équations aux dérivées partielles (2012): 1-10. <http://eudml.org/doc/275601>.

@article{Cacciafesta2012,
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.},
affiliation = {SAPIENZA — Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy},
author = {Cacciafesta, Federico},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
pages = {1-10},
publisher = {Groupement de recherche 2434 du CNRS},
title = {The cubic nonlinear Dirac equation},
url = {http://eudml.org/doc/275601},
year = {2012},
}

TY - JOUR
AU - Cacciafesta, Federico
TI - The cubic nonlinear Dirac equation
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 10
AB - 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.
LA - eng
UR - http://eudml.org/doc/275601
ER -

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. Zbl1230.35111MR2833377
  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. Zbl1260.35159
  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. Zbl0637.35014MR916298
  4. Miguel Escobedo and Luis Vega. A semilinear Dirac equation in H s ( R 3 ) for s &gt; 1 . SIAM J. Math. Anal., 28(2):338–362, 1997. Zbl0877.35028MR1434039
  5. Daoyuan Fang and Chengbo Wang. Some remarks on Strichartz estimates for homogeneous wave equation. Nonlinear Anal., 65(3):697–706, 2006. Zbl1096.35026MR2231083
  6. Daoyuan Fang and Chengbo Wang. Weighted Strichartz estimates with angular regularity and their applications. 2008. Zbl1226.35008
  7. Chengbo Wang Jin-Cheng Jiang and Xin Yu. Generalized and weighted strichartz estimates. 2010. Zbl1264.35011
  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. Zbl0803.35095MR1231427
  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. Zbl1060.35025MR2108356
  10. Shuji Machihara, Makoto Nakamura, and Tohru Ozawa. Small global solutions for nonlinear Dirac equations. Differential Integral Equations, 17(5-6):623–636, 2004. Zbl1174.35452MR2054938
  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). Zbl0734.35097MR1080773
  12. Branko Najman. The nonrelativistic limit of the nonlinear Dirac equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 9(1):3–12, 1992. Zbl0746.35036MR1151464
  13. Michael Reed. Abstract non-linear wave equations. Lecture Notes in Mathematics, Vol. 507. Springer-Verlag, Berlin, 1976. Zbl0317.35002MR605679
  14. Jacob Sterbenz Angular regularity and Strichartz estimates for the wave equation. Int. Math. Res. Not. 2005, no. 4, 187Ð231. Zbl1072.35048MR2128434
  15. Bernd Thaller. The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. Zbl0765.47023MR1219537

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