# 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 and 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 and Continuous Dynamical Systems, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123 [2] Ruoci Sun. Filtering the $L^2-$critical focusing Schrödinger equation. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5973-5990. doi: 10.3934/dcds.2020255 [3] Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $n$. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030 [4] Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $l(s^2)$. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056 [5] Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086 [6] Liejun Shen, Marco Squassina, Minbo Yang. Critical gauged Schrödinger equations in $\mathbb{R}^2$ with vanishing potentials. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022059 [7] Yong Luo, Shu Zhang. Concentration behavior of ground states for $L^2$-critical Schrödinger Equation with a spatially decaying nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1481-1504. doi: 10.3934/cpaa.2022026 [8] Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $L^p$-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028 [9] 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 and Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [10] Jiangang Qi, Bing Xie. Extremum estimates of the $L^1$-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243 [11] Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $L^2$-subcritical data. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034 [12] Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $L^2$-norm control constraint. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022030 [13] Dongyi Liu, Genqi Xu. Input-output $L^2$-well-posedness, regularity and Lyapunov stability of string equations on networks. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022007 [14] Sawsan Alhowaity, Ernesto Pérez-Chavela, Juan Manuel Sánchez-Cerritos. The curved symmetric $2$– and $3$–center problem on constant negative surfaces. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2941-2963. doi: 10.3934/cpaa.2021090 [15] Peili Li, Xiliang Lu, Yunhai Xiao. Smoothing Newton method for $\ell^0$-$\ell^2$ regularized linear inverse problem. Inverse Problems and Imaging, 2022, 16 (1) : 153-177. doi: 10.3934/ipi.2021044 [16] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [17] 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 and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [18] Xuerui Gao, Yanqin Bai, Shu-Cherng Fang, Jian Luo, Qian Li. A new hybrid $l_p$-$l_2$ model for sparse solutions with applications to image processing. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021211 [19] Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072 [20] Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $L^{2}$ initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011

2020 Impact Factor: 1.081