Local well-posedness for the nonlinear Dirac equation in two space dimensions

Hartmut Pecher

doi:10.3934/cpaa.2014.13.673

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2014, Volume 13, Issue 2: 673-685. Doi: 10.3934/cpaa.2014.13.673

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Local well-posedness for the nonlinear Dirac equation in two space dimensions

Hartmut Pecher^1,

1.
Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42097 Wuppertal

Received: February 28, 2013

Revised: July 31, 2013

Published: February 2014

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Abstract

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in $H^s$ for $ s > 1/2$. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.

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Mathematics Subject Classification: 35Q55, 35L70.

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References

[1]	P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, Journal of the EMS, 9 (2007), 877-898.
[2]	P. d'Ancona, D. Foschi and S. Selberg, Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions, Journal Hyperbolic Diff. Equations, 4 (2007), 295-330.
[3]	T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.
[4]	V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension, Proc. AMS, 69 (1978), 289-296.
[5]	M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s(R^3)$ for $s>1$, SIAM J. Math. Anal., 28 (1997), 338-362.
[6]	R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Phys. Rev., 103 (1956), 1571-1579.
[7]	R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Phys. Rev., 83 (1951), 326-333.
[8]	D. Foschi and S. Klainerman, Homogeneous $L^2$ bilinear estimates for wave equations, Ann. Scient. ENS $4^e$ serie, 23 (2000), 211-274.
[9]	J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Analysis, 133 (1995), 50-68.
[10]	D. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D, 10 (1974), 3235-3253.
[11]	A. Grünrock and H. Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions, Comm. Partial Differential Equations, 35 (2010), 89-112.
[12]	S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219 (2005), 1-20.
[13]	S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation, Rev. Math. Iberoamericana, 19 (2003), 179-194.
[14]	S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Ph.D. thesis, Princeton Univ., 1999.
[15]	S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Diff. Int. Equ., 23 (2010), 265-278.
[16]	M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.
[17]	W.E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112.