May  2018, 38(5): 2555-2569. doi: 10.3934/dcds.2018107

Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field

Department of Mathematics, University of Bergen, PO Box 7803, 5020 Bergen, Norway

Received  July 2017 Published  March 2018

Fund Project: The author thanks Jean-Christophe Merle for his hospitality during the author's visit to the University of Vechta, where the main part of the research reported here was carried out. The author also thanks the referee for many useful comments and suggestions.

We prove global well-posedness for the coupled Maxwell-Dirac-Thirring-Gross-Neveu equations in one space dimension, with data for the Dirac spinor in the critical space $L^2(\mathbb{R})$. In particular, we recover earlier results of Candy and Huh for the Thirring and Gross-Neveu models, respectively, without the coupling to the electromagnetic field, but the function spaces we introduce allow for a greatly simplified proof. We also apply our method to prove local well-posedness in $L^2(\mathbb{R})$ for a quadratic Dirac equation, improving an earlier result of Tesfahun and the author.

Citation: Sigmund Selberg. Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2555-2569. doi: 10.3934/dcds.2018107
References:
[1]

A. Bachelot, Global Cauchy problem for semilinear hyperbolic systems with nonlocal interactions. Applications to Dirac equations, J. Math. Pures Appl.(9), 86 (2006), 201-236.  doi: 10.1016/j.matpur.2006.01.006.

[2]

I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\Bbb{R}^3)$, Comm. Math. Phys., 335 (2015), 43-82.  doi: 10.1007/s00220-014-2164-0.

[3]

N. Bournaveas, A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension, J. Funct. Anal., 173 (2000), 203-213.  doi: 10.1006/jfan.1999.3559.

[4]

N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension, Discrete Contin. Dyn. Syst., 20 (2008), 605-616. 

[5]

N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not. IMRN, (2016), 6735-6828. 

[6]

N. Boussaïd and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524.  doi: 10.1016/j.jfa.2016.04.013.

[7]

T. Candy and H. Lindblad, Long Range Scattering for the cubic Dirac Equation on $\mathbf{R}^{1+1}$, arXiv e-prints 1606.08397 (2016).

[8]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. 

[9]

_______, Bilinear estimates and applications to global well-posedness for the Dirac-KleinGordon equation on $\Bbb R^{1+1}$, J. Hyperbolic Differ. Equ., 10 (2013), 1-35. doi: 10.1142/S021989161350001X.

[10]

J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension, J. Functional Analysis, 13 (1973), 173-184.  doi: 10.1016/0022-1236(73)90043-8.

[11]

A. ContrerasD. E. Pelinovsky and Y. Shimabukuro, $L^2$ orbital stability of Dirac solitons in the massive Thirring model, Comm. Partial Differential Equations, 41 (2016), 227-255.  doi: 10.1080/03605302.2015.1123272.

[12]

P. D'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. 

[13]

_______, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839. doi: 10.1353/ajm.0.0118.

[14]

P. D'Ancona and S. Selberg, Global well-posedness of the Maxwell-Dirac system in two space dimensions, J. Funct. Anal., 260 (2011), 2300-2365.  doi: 10.1016/j.jfa.2010.12.010.

[15]

V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension, Proc. Amer. Math. Soc., 69 (1978), 289-296.  doi: 10.1090/S0002-9939-1978-0463658-5.

[16]

J.-P. Dias and M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ricerche Mat., 35 (1986), 309-316. 

[17]

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. 

[18]

H. Huh, Global charge solutions of Maxwell-Dirac equations in $\Bbb R^{1+1}$, J. Phys. A, 43 (2010), 445206, 7pp. doi: 10.1088/1751-8113/43/44/445206.

[19]

_______, Global strong solution to the Thirring model in critical space, J. Math. Anal. Appl., 381 (2011), 513-520. doi: 10.1016/j.jmaa.2011.02.042.

[20]

_______, Global solutions to Gross-Neveu equation, Lett. Math. Phys., 103 (2013), 927-931. doi: 10.1007/s11005-013-0622-9.

[21]

H. Huh and B. Moon, Low regularity well-posedness for Gross-Neveu equations, Commun. Pure Appl. Anal., 14 (2015), 1903-1913.  doi: 10.3934/cpaa.2015.14.1903.

[22]

M. Ikeda, Final state problem for the Dirac-Klein-Gordon equations in two space dimensions, Abstr. Appl. Anal., (2013), Art. ID 273959, 11pp.

[23]

S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst., 13 (2005), 277-290.  doi: 10.3934/dcds.2005.13.277.

[24]

_______, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484.

[25]

S. MachiharaK. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math., 50 (2010), 403-451.  doi: 10.1215/0023608X-2009-018.

[26]

S. Machihara and M. Okamoto, Remarks on ill-posedness for the Dirac-Klein-Gordon system, Dyn. Partial Differ. Equ., 13 (2016), 179-190.  doi: 10.4310/DPDE.2016.v13.n3.a1.

[27]

I. P. Naumkin, Cubic nonlinear Dirac equation in a quarter plane, J. Math. Anal. Appl., 434 (2016), 1633-1664.  doi: 10.1016/j.jmaa.2015.09.049.

[28]

_______, Initial-boundary value problem for the one dimensional Thirring model, J. Differential Equations, 261 (2016), 4486-4523. doi: 10.1016/j.jde.2016.07.003.

[29]

M. Okamoto, Well-posedness and ill-posedness of the Cauchy problem for the Maxwell-Dirac system in $1+1$ space time dimensions, Adv. Differential Equations, 18 (2013), 179-199. 

[30]

H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. 

[31]

S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations, 23 (2010), 265-278. 

[32]

A. Tesfahun, Global well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index, J. Hyperbolic Differ. Equ., 6 (2009), 631-661.  doi: 10.1142/S0219891609001952.

[33]

X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not. IMRN, (2015), 10801-10846. 

[34]

A. You and Y. Zhang, Global solution to Maxwell-Dirac equations in $1+1$ dimensions, Nonlinear Anal., 98 (2014), 226-236.  doi: 10.1016/j.na.2013.12.014.

[35]

Y. Zhang and Q. Zhao, Global solution to nonlinear Dirac equation for Gross-Neveu model in $1+1$ dimensions, Nonlinear Anal., 118 (2015), 82-96.  doi: 10.1016/j.na.2015.02.007.

show all references

References:
[1]

A. Bachelot, Global Cauchy problem for semilinear hyperbolic systems with nonlocal interactions. Applications to Dirac equations, J. Math. Pures Appl.(9), 86 (2006), 201-236.  doi: 10.1016/j.matpur.2006.01.006.

[2]

I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\Bbb{R}^3)$, Comm. Math. Phys., 335 (2015), 43-82.  doi: 10.1007/s00220-014-2164-0.

[3]

N. Bournaveas, A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension, J. Funct. Anal., 173 (2000), 203-213.  doi: 10.1006/jfan.1999.3559.

[4]

N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension, Discrete Contin. Dyn. Syst., 20 (2008), 605-616. 

[5]

N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not. IMRN, (2016), 6735-6828. 

[6]

N. Boussaïd and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524.  doi: 10.1016/j.jfa.2016.04.013.

[7]

T. Candy and H. Lindblad, Long Range Scattering for the cubic Dirac Equation on $\mathbf{R}^{1+1}$, arXiv e-prints 1606.08397 (2016).

[8]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. 

[9]

_______, Bilinear estimates and applications to global well-posedness for the Dirac-KleinGordon equation on $\Bbb R^{1+1}$, J. Hyperbolic Differ. Equ., 10 (2013), 1-35. doi: 10.1142/S021989161350001X.

[10]

J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension, J. Functional Analysis, 13 (1973), 173-184.  doi: 10.1016/0022-1236(73)90043-8.

[11]

A. ContrerasD. E. Pelinovsky and Y. Shimabukuro, $L^2$ orbital stability of Dirac solitons in the massive Thirring model, Comm. Partial Differential Equations, 41 (2016), 227-255.  doi: 10.1080/03605302.2015.1123272.

[12]

P. D'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. 

[13]

_______, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839. doi: 10.1353/ajm.0.0118.

[14]

P. D'Ancona and S. Selberg, Global well-posedness of the Maxwell-Dirac system in two space dimensions, J. Funct. Anal., 260 (2011), 2300-2365.  doi: 10.1016/j.jfa.2010.12.010.

[15]

V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension, Proc. Amer. Math. Soc., 69 (1978), 289-296.  doi: 10.1090/S0002-9939-1978-0463658-5.

[16]

J.-P. Dias and M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ricerche Mat., 35 (1986), 309-316. 

[17]

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. 

[18]

H. Huh, Global charge solutions of Maxwell-Dirac equations in $\Bbb R^{1+1}$, J. Phys. A, 43 (2010), 445206, 7pp. doi: 10.1088/1751-8113/43/44/445206.

[19]

_______, Global strong solution to the Thirring model in critical space, J. Math. Anal. Appl., 381 (2011), 513-520. doi: 10.1016/j.jmaa.2011.02.042.

[20]

_______, Global solutions to Gross-Neveu equation, Lett. Math. Phys., 103 (2013), 927-931. doi: 10.1007/s11005-013-0622-9.

[21]

H. Huh and B. Moon, Low regularity well-posedness for Gross-Neveu equations, Commun. Pure Appl. Anal., 14 (2015), 1903-1913.  doi: 10.3934/cpaa.2015.14.1903.

[22]

M. Ikeda, Final state problem for the Dirac-Klein-Gordon equations in two space dimensions, Abstr. Appl. Anal., (2013), Art. ID 273959, 11pp.

[23]

S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst., 13 (2005), 277-290.  doi: 10.3934/dcds.2005.13.277.

[24]

_______, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484.

[25]

S. MachiharaK. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math., 50 (2010), 403-451.  doi: 10.1215/0023608X-2009-018.

[26]

S. Machihara and M. Okamoto, Remarks on ill-posedness for the Dirac-Klein-Gordon system, Dyn. Partial Differ. Equ., 13 (2016), 179-190.  doi: 10.4310/DPDE.2016.v13.n3.a1.

[27]

I. P. Naumkin, Cubic nonlinear Dirac equation in a quarter plane, J. Math. Anal. Appl., 434 (2016), 1633-1664.  doi: 10.1016/j.jmaa.2015.09.049.

[28]

_______, Initial-boundary value problem for the one dimensional Thirring model, J. Differential Equations, 261 (2016), 4486-4523. doi: 10.1016/j.jde.2016.07.003.

[29]

M. Okamoto, Well-posedness and ill-posedness of the Cauchy problem for the Maxwell-Dirac system in $1+1$ space time dimensions, Adv. Differential Equations, 18 (2013), 179-199. 

[30]

H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. 

[31]

S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations, 23 (2010), 265-278. 

[32]

A. Tesfahun, Global well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index, J. Hyperbolic Differ. Equ., 6 (2009), 631-661.  doi: 10.1142/S0219891609001952.

[33]

X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not. IMRN, (2015), 10801-10846. 

[34]

A. You and Y. Zhang, Global solution to Maxwell-Dirac equations in $1+1$ dimensions, Nonlinear Anal., 98 (2014), 226-236.  doi: 10.1016/j.na.2013.12.014.

[35]

Y. Zhang and Q. Zhao, Global solution to nonlinear Dirac equation for Gross-Neveu model in $1+1$ dimensions, Nonlinear Anal., 118 (2015), 82-96.  doi: 10.1016/j.na.2015.02.007.

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