The cubic nonlinear Dirac equation

Cacciafesta, Federico

Journées équations aux dérivées partielles (2012), article no. 1, 10 p.

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.

[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 $H^s\left({\mathbf{R}}^3\right)$ 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