# American Institute of Mathematical Sciences

September  2015, 14(5): 1903-1913. doi: 10.3934/cpaa.2015.14.1903

## Low regularity well-posedness for Gross-Neveu equations

 1 Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea

Received  October 2014 Revised  February 2015 Published  June 2015

We address the problem of local and global well-posedness of Gross-Neveu (GN) equations for low regularity initial data. Combined with the standard machinery of $X_R$, $Y_R$ and $X^{s,b}$ spaces, we obtain local-wellposedness of (GN) for initial data $u, v \in H^s$ with $s\geq 0$. To prove the existence of global solution for the critical space $L^2$, we show non concentration of $L^2$ norm.
Citation: Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903
##### References:
 [1] T. Candy, Global existence for $L^2$ critical nonlinear Dirac equation in one dimension,, \emph{Adv. Differential Equations}, 16 (2011), 643. Google Scholar [2] V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension,, \emph{Proc. Amer. Math. Soc.}, 69 (1978), 289. Google Scholar [3] D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, \emph{Phys. Rev. D}, 10 (1974), 3235. Google Scholar [4] W. E. Thirring, A soluble relativistic field theory,, \emph{Ann.Phys.}, 3 (1958), 91. Google Scholar [5] H. Huh, Global strong solution to the Thirring model in critical space,, \emph{J. Math. Anal. Appl.}, 381 (2011), 513. doi: 10.1016/j.jmaa.2011.02.042. Google Scholar [6] H. Huh, Global solutions to Gross-Neveu equations,, \emph{Lett. Math. Phys.}, 103 (2013), 927. doi: 10.1007/s11005-013-0622-9. Google Scholar [7] S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension,, \emph{Kyoto J. Math.}, 50 (2010), 403. doi: 10.1215/0023608X-2009-018. Google Scholar [8] S. Selberg and A. Tefahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension,, \emph{Differential and Integral Equations}, 23 (2010), 265. Google Scholar [9] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system,, \emph{Journal of EMS}, 9 (2007), 877. doi: 10.4171/JEMS/100. Google Scholar [10] Y. Zhang, Global strong solution to a nonlinear Dirac type equation in one dimension,, \emph{Nonlinear Analysis}, 80 (2013), 150. doi: 10.1016/j.na.2012.10.008. Google Scholar

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##### References:
 [1] T. Candy, Global existence for $L^2$ critical nonlinear Dirac equation in one dimension,, \emph{Adv. Differential Equations}, 16 (2011), 643. Google Scholar [2] V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension,, \emph{Proc. Amer. Math. Soc.}, 69 (1978), 289. Google Scholar [3] D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, \emph{Phys. Rev. D}, 10 (1974), 3235. Google Scholar [4] W. E. Thirring, A soluble relativistic field theory,, \emph{Ann.Phys.}, 3 (1958), 91. Google Scholar [5] H. Huh, Global strong solution to the Thirring model in critical space,, \emph{J. Math. Anal. Appl.}, 381 (2011), 513. doi: 10.1016/j.jmaa.2011.02.042. Google Scholar [6] H. Huh, Global solutions to Gross-Neveu equations,, \emph{Lett. Math. Phys.}, 103 (2013), 927. doi: 10.1007/s11005-013-0622-9. Google Scholar [7] S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension,, \emph{Kyoto J. Math.}, 50 (2010), 403. doi: 10.1215/0023608X-2009-018. Google Scholar [8] S. Selberg and A. Tefahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension,, \emph{Differential and Integral Equations}, 23 (2010), 265. Google Scholar [9] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system,, \emph{Journal of EMS}, 9 (2007), 877. doi: 10.4171/JEMS/100. Google Scholar [10] Y. Zhang, Global strong solution to a nonlinear Dirac type equation in one dimension,, \emph{Nonlinear Analysis}, 80 (2013), 150. doi: 10.1016/j.na.2012.10.008. Google Scholar
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