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Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness
Department of Mathematics, Chung-Ang University, Seoul, 156-756, Korea |
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.
References:
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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. |
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T. Candy,
Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.
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M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/math/0503366. |
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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. |
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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. |
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H. Huh,
Global strong solution to the Thirring model in critical space, J. Math. Anal. Appl., 381 (2011), 513-520.
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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. |
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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.
|
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