# 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.

show all references

##### 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.
 [1] Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $L^2$-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123 [2] Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $l(s^2)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056 [3] Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086 [4] Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [5] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [6] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [7] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [8] Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $\mathbb{T}^3$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116 [9] Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012 [10] Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094 [11] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [12] Jayadev S. Athreya, Yitwah Cheung, Howard Masur. Siegel–Veech transforms are in $\boldsymbol{L^2}$(with an appendix by Jayadev S. Athreya and Rene Rühr). Journal of Modern Dynamics, 2019, 14: 1-19. doi: 10.3934/jmd.2019001 [13] Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129 [14] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [15] Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $\mathbb{H}^2$ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189 [16] Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n}$. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 [17] Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $\mathbb S^2$ and $\mathbb H^2$ are inclined. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020067 [18] Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131 [19] Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038 [20] Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

2017 Impact Factor: 1.049