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http://dx.doi.org/10.4134/BKMS.b201040

LOCAL WELL-POSEDNESS OF DIRAC EQUATIONS WITH NONLINEARITY DERIVED FROM HONEYCOMB STRUCTURE IN 2 DIMENSIONS  

Lee, Kiyeon (Department of Mathematics Ewha Womans University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1445-1461 More about this Journal
Abstract
The aim of this paper is to show the local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlin-earity derived from honeycomb structure in Hs for s > $\frac{7}{8}$ and s > $\frac{3}{8}$, respectively. We also provide the smoothness failure of flows of Dirac equations.
Keywords
Dirac equations; honeycomb lattice; local well-posedness; non-smoothness; Bourgain's space;
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1 T. Tao, Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839-908.   DOI
2 A. Tesfahun, Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity in ℝ1+2, Int. Math. Res. Not. IMRN 2020 (2020), no. 19, 6489-6538. https://doi.org/10.1093/imrn/rny217   DOI
3 C. Yang, Scattering results for Dirac Hartree-type equations with small initial data, Commun. Pure Appl. Anal. 18 (2019), no. 4, 1711-1734. https://doi.org/10.3934/cpaa.2019081   DOI
4 Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal. 38 (2006), no. 4, 1060-1074. https://doi.org/10.1137/060653688   DOI
5 J. Arbunich and C. Sparber, Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures, J. Math. Phys. 59 (2018), no. 1, 011509, 18 pp. https://doi.org/10.1063/1.5021754   DOI
6 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrodinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107-156. https://doi.org/10.1007/BF01896020   DOI
7 O. Bahat-Treidel, O. Peleg, and M. Segev, Symmetry breaking in honeycomb photonic lattices, Opt. Lett. 33 (2008), 2251-2253.   DOI
8 I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^{\frac{1}{2}}$ (ℝ2), Comm. Math. Phys. 343 (2016), no. 2, 515-562. https://doi.org/10.1007/s00220-015-2508-4   DOI
9 I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrodinger equation, J. Funct. Anal. 233 (2006), no. 1, 228-259. https://doi.org/10.1016/j.jfa.2005.08.004   DOI
10 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009), 109-162.   DOI
11 Y. Cho and K. Lee, Small data scattering of Dirac equations with Yukawa type potentials in L2x(ℝ2), Diff. Inte. Equ. 34 (7/8) (2021), 425-436.
12 Y. Cho, K. Lee, and T. Ozawa, Small data scattering of 2d Hatree type Dirac equations, J. Math. Anal. Appl. 506 (2022), no. 1, 125549. https://doi.org/10.1016/j.jmaa.2021.125549   DOI
13 Y. Cho, T. Ozawa, H. Sasaki, and Y. Shim, Remarks on the semirelativistic Hartree equations, Discrete Contin. Dyn. Syst. 23 (2009), no. 4, 1277-1294. https://doi.org/10.3934/dcds.2009.23.1277   DOI
14 V. D. Dinh, On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces, Discrete Contin. Dyn. Syst. 38 (2018), no. 3, 1127-1143. https://doi.org/10.3934/dcds.2018047   DOI
15 K. Fujiwara, V. Georgiev, and T. Ozawa, Higher order fractional Leibniz rule, J. Fourier Anal. Appl. 24 (2018), no. 3, 650-665. https://doi.org/10.1007/s00041-017-9541-y   DOI
16 K. Fujiwara, V. Georgiev, and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, J. Math. Pures Appl. (9) 136 (2020), 239-256. https://doi.org/10.1016/j.matpur.2019.10.003   DOI
17 K. Lee, Low regularity well-posedness of Hartree type Dirac equations in 2,3-dimensions, To appear in Comm. Pure. Appl. Anal. https://doi.org/10.3934/cpaa.2021126   DOI
18 E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom. 10 (2007), no. 1, 43-64. https://doi.org/10.1007/s11040-007-9020-9   DOI
19 J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384-436. https://doi.org/10.1006/jfan.1997.3148   DOI
20 R. El Hajj and F. Mehats, Analysis of models for quantum transport of electrons in graphene layers, Math. Models Methods Appl. Sci. 24 (2014), no. 11, 2287-2310. https://doi.org/10.1142/S0218202514500213   DOI
21 C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573-603. https://doi.org/10.1090/S0894-0347-96-00200-7   DOI
22 T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. https://doi.org/10.1002/cpa.3160410704   DOI
23 C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527-620. https://doi.org/10.1002/cpa.3160460405   DOI
24 S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal. 97 (2014), 125-137. https://doi.org/10.1016/j.na.2013.11.023   DOI
25 S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Differential Equations 16 (2011), no. 7-8, 667-690.   DOI
26 A. Tesfahun, Small data scattering for cubic Dirac equation with Hartree type nonlinearity in ℝ1+3, SIAM J. Math. Anal. 52 (2020), no. 3, 2969-3003. https://doi.org/10.1137/17M1155788   DOI
27 L. Molinet, J. C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), no. 4, 982-988. https://doi.org/10.1137/S0036141001385307   DOI