# American Institute of Mathematical Sciences

March  2014, 13(2): 673-685. doi: 10.3934/cpaa.2014.13.673

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

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

Received  March 2013 Revised  August 2013 Published  October 2013

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.
Citation: Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
##### 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.   Google Scholar [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.   Google Scholar [3] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension,, Adv. Differential Equations, 16 (2011), 643.   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field,, Phys. Rev., 103 (1956), 1571.   Google Scholar [7] R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields,, Phys. Rev., 83 (1951), 326.   Google Scholar [8] D. Foschi and S. Klainerman, Homogeneous $L^2$ bilinear estimates for wave equations,, Ann. Scient. ENS $4^e$ serie, 23 (2000), 211.   Google Scholar [9] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Analysis, 133 (1995), 50.   Google Scholar [10] D. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, Phys. Rev. D, 10 (1974), 3235.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999).   Google Scholar [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.   Google Scholar [16] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy,, Phys. Rev. D, 1 (1970), 2766.   Google Scholar [17] W.E. Thirring, A soluble relativistic field theory,, Ann. Physics, 3 (1958), 91.   Google Scholar

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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.   Google Scholar [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.   Google Scholar [3] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension,, Adv. Differential Equations, 16 (2011), 643.   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field,, Phys. Rev., 103 (1956), 1571.   Google Scholar [7] R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields,, Phys. Rev., 83 (1951), 326.   Google Scholar [8] D. Foschi and S. Klainerman, Homogeneous $L^2$ bilinear estimates for wave equations,, Ann. Scient. ENS $4^e$ serie, 23 (2000), 211.   Google Scholar [9] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Analysis, 133 (1995), 50.   Google Scholar [10] D. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, Phys. Rev. D, 10 (1974), 3235.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999).   Google Scholar [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.   Google Scholar [16] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy,, Phys. Rev. D, 1 (1970), 2766.   Google Scholar [17] W.E. Thirring, A soluble relativistic field theory,, Ann. Physics, 3 (1958), 91.   Google Scholar
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