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.  Google Scholar

[2]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.   Google Scholar

[3]

M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/math/0503366. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[2]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.   Google Scholar

[3]

M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/math/0503366. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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]

Ruoci Sun. Filtering the $ L^2- $critical focusing Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5973-5990. doi: 10.3934/dcds.2020255

[3]

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

[4]

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

[5]

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

[6]

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 & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020243

[7]

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

[8]

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

[9]

Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011

[10]

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

[11]

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

[12]

Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061

[13]

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

[14]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020077

[15]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020298

[16]

Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121

[17]

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

[18]

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

[19]

Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239

[20]

Xin-Guang Yang, Marcelo J. D. Nascimento, Maurício L. Pelicer. Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (164)
  • HTML views (400)
  • Cited by (0)

Other articles
by authors

[Back to Top]