과제정보
K. Lee was partially supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2020R1F1A1A0106876811).
참고문헌
- 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
- O. Bahat-Treidel, O. Peleg, and M. Segev, Symmetry breaking in honeycomb photonic lattices, Opt. Lett. 33 (2008), 2251-2253. https://doi.org/10.1364/OL.33.002251
-
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 - 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
- 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
- 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. https://doi.org/10.1103/RevModPhys.81.109
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Differential Equations 16 (2011), no. 7-8, 667-690. https://doi.org/10.57262/ade/1355703202
- T. Tao, Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839-908. https://doi.org/10.1353/ajm.2001.0035
- 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
- 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
- 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