# American Institute of Mathematical Sciences

March 2018, 7(1): 53-60. doi: 10.3934/eect.2018003

## Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness

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

Received  August 2017 Revised  November 2017 Published  January 2018

Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space $C([0, T]; H^{s}(\Bbb{R}))$ $(s<0)$ to the system (1) which is $L^2(\Bbb{R})$ scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.

Citation: Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
##### References:
 [1] D. Agueev and D. Pelinovsky, Modeling of wave resonances in low-contrast photonic crystals, SIAM J. Appl. Math., 65 (2005), 1101-1129. doi: 10.1137/040606053. [2] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. [3] M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/math/0503366. [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. Amer. Math. Soc., 69 (1978), 289-296. doi: 10.1090/S0002-9939-1978-0463658-5. [5] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal., 27 (1996), 708-724. doi: 10.1137/0527038. [6] H. Huh, 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. [7] H. Huh, Remarks on nonlinear Dirac equations in one space dimension, Commun. Korean Math. Soc. Soc., 30 (2015), 201-208. doi: 10.4134/CKMS.2015.30.3.201. [8] S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential and Integral Equations, 23 (2010), 265-278.

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##### References:
 [1] D. Agueev and D. Pelinovsky, Modeling of wave resonances in low-contrast photonic crystals, SIAM J. Appl. Math., 65 (2005), 1101-1129. doi: 10.1137/040606053. [2] T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. [3] M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/math/0503366. [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. Amer. Math. Soc., 69 (1978), 289-296. doi: 10.1090/S0002-9939-1978-0463658-5. [5] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal., 27 (1996), 708-724. doi: 10.1137/0527038. [6] H. Huh, 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. [7] H. Huh, Remarks on nonlinear Dirac equations in one space dimension, Commun. Korean Math. Soc. Soc., 30 (2015), 201-208. doi: 10.4134/CKMS.2015.30.3.201. [8] S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential and Integral Equations, 23 (2010), 265-278.
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