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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

Hartmut Pecher 1,

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.
Keywords: Dirac equation, well-posedness, Fourier restriction norm method..
Mathematics Subject Classification: 35Q55, 35L70.
Citation: Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and 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), 877-898.

[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.

show all references

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.

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