# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2018, 38 (5) : 2555-2569. doi: 10.3934/dcds.2018107
##### References:
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Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524. doi: 10.1016/j.jfa.2016.04.013. Google Scholar [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).Google Scholar [8] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. Google Scholar [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. Google Scholar [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. Google Scholar [11] A. Contreras, D. 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. Google Scholar [12] P. D'Ancona, D. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] _______, Global solutions to Gross-Neveu equation, Lett. Math. Phys., 103 (2013), 927-931. doi: 10.1007/s11005-013-0622-9. Google Scholar [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. Google Scholar [22] M. Ikeda, Final state problem for the Dirac-Klein-Gordon equations in two space dimensions, Abstr. Appl. Anal., (2013), Art. ID 273959, 11pp. Google Scholar [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. Google Scholar [24] _______, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484. Google Scholar [25] S. Machihara, K. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. Google Scholar [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. Google Scholar [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. Google Scholar [33] X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not. IMRN, (2015), 10801-10846. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not. IMRN, (2016), 6735-6828. Google Scholar [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. Google Scholar [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).Google Scholar [8] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. Google Scholar [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. Google Scholar [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. Google Scholar [11] A. Contreras, D. 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. Google Scholar [12] P. D'Ancona, D. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] _______, Global solutions to Gross-Neveu equation, Lett. Math. Phys., 103 (2013), 927-931. doi: 10.1007/s11005-013-0622-9. Google Scholar [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. Google Scholar [22] M. Ikeda, Final state problem for the Dirac-Klein-Gordon equations in two space dimensions, Abstr. Appl. Anal., (2013), Art. ID 273959, 11pp. Google Scholar [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. Google Scholar [24] _______, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484. Google Scholar [25] S. Machihara, K. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. Google Scholar [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. Google Scholar [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. Google Scholar [33] X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not. IMRN, (2015), 10801-10846. Google Scholar [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. Google Scholar [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. Google Scholar
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