Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

LOW REGULARITY SOLUTIONS TO HIGHER-ORDER HARTREE-FOCK EQUATIONS WITH UNIFORM BOUNDS

  • Changhun Yang (Department of Mathematics Chungbuk National University)
  • Received : 2024.01.02
  • Accepted : 2024.01.23
  • Published : 2024.02.29
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the higher-order HartreeFock equations. The higher-order linear Schrödinger equation was introduced in [5] as the formal finite Taylor expansion of the pseudorelativistic linear Schrödinger equation. In [13], the authors established global-in-time Strichartz estimates for the linear higher-order equations which hold uniformly in the speed of light c ≥ 1 and as their applications they proved the convergence of higher-order Hartree-Fock equations to the corresponding pseudo-relativistic equation on arbitrary time interval as c goes to infinity when the Taylor expansion order is odd. To achieve this, they not only showed the existence of solutions in L2 space but also proved that the solutions stay bounded uniformly in c. We address the remaining question on the convergence of higherorder Hartree-Fock equations when the Taylor expansion order is even. The distinguished feature from the odd case is that the group velocity of phase function would be vanishing when the size of frequency is comparable to c. Owing to this property, the kinetic energy of solutions is not coercive and only weaker Strichartz estimates compared to the odd case were obtained in [13]. Thus, we only manage to establish the existence of local solutions in Hs space for s > $\frac{1}{3}$ on a finite time interval [-T, T], however, the time interval does not depend on c and the solutions are bounded uniformly in c. In addition, we provide the convergence result of higher-order Hartree-Fock equations to the pseudo-relativistic equation with the same convergence rate as the odd case, which holds on [-T, T].

Keywords

Acknowledgement

The author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1C1C1005700).

References

  1. P. Bechouche, N. J. Mauser, and S. Selberg, Nonrelativistic limit of KleinGordon-Maxwell to Schrodinger-Poisson, Amer. J. Math., 126 (2004), no. 1, 31-64. https://doi.org/10.1353/ajm.2004.0001
  2. P. Bechouche, N. J. Mauser, and S. Selberg, On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit, J. Hyperbolic Differ. Equ., 2 (2005), no. 1, 129-182. https://doi.org/10.1142/S0219891605000415
  3. N. Benedikter, M. Porta, and B. Schlein, Mean-field dynamics of fermions with relativistic dispersion, J. Math. Phys., 55, no. 2, 021901.
  4. R. Carles, W. Lucha, and E. Moulay, Higher-order schrodinger and hartree-fock equations, J. Math. Phys., 56 (2015), no. 12, 122301.
  5. R. Carles, and E. Moulay, Higher order schrodinger equations, J. Phys. A., 45 (2012), no. 39, 395304.
  6. Y. Cho, Short-range scattering of Hartree type fractional NLS, J. Differ. Equ., 262 (2017), no. 1, 116-144. https://doi.org/10.1016/j.jde.2016.09.025
  7. 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
  8. W. Choi, Y. Hong, and J. Seok, Optimal convergence rate and regularity of nonrelativistic limit for the nonlinear pseudo-relativistic equations, J. Funct. Anal., 274 (2018), no. 3, 695-722. https://doi.org/10.1016/j.jfa.2017.11.006
  9. W. Choi, Y. Hong, and J. Seok, Uniqueness and symmetry of ground states for higher-order equations, Cal. Var. PDE., 57 (2018), no. 3, 77.
  10. W. Choi, and J. Seok, Nonrelativistic limit of standing waves for pseudorelativistic nonlinear Schrodinger equations, J. Math. Phys., 57 (2016), no. 2, 021510, 15.
  11. J. Frohlich and E. Lenzmann, Dynamical collapse of white dwarfs in Hartreeand Hartree-Fock theory, Comm. Math. Phys., 274 (2007), no. 3, 737-750. https://doi.org/10.1007/s00220-007-0290-7
  12. 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
  13. Y. Hong, C. Kwak, and C. Yang, Strichartz estimates for higher-order schrodinger equations and their applications, J. Differ. Equ., 324 (2022), 41-75. https://doi.org/10.1016/j.jde.2022.03.040
  14. 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
  15. E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal., PDE., 2 (2009), no. 1, 1-27. https://doi.org/10.2140/apde.2009.2.1
  16. E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics., 155 (1984), no. 2, 494-512. https://doi.org/10.1016/0003-4916(84)90010-1
  17. E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), no. 1, 147-174. https://doi.org/10.1007/BF01217684
  18. S. Machihara, K. Nakanishi, and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear klein-gordon equations, Math. Ann., 322 (2002), no. 3, 603-621. https://doi.org/10.1007/s002080200008
  19. S. Machihara, K. Nakanishi, and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoamericana 19 (2003), no. 1, 179-194. https://doi.org/10.4171/rmi/342
  20. N. Masmoudi and K. Nakanishi, From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrodinger equations, Math. Ann., 324 (2002), no. 2, 359-389. https://doi.org/10.1007/s00208-002-0342-4
  21. N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-KleinGordon and Maxwell-Dirac to Poisson-Schrodinger, Int. Math. Res. Not. (2003), no. 13, 697-734.
  22. N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to the nonlinear Schrodinger equation, J. Hyperbolic Differ. Equ., 2 (2005), no. 4, 975-1008. https://doi.org/10.1142/S0219891605000683