Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GLOBAL SOLUTIONS OF SEMIRELATIVISTIC HARTREE TYPE EQUATIONS

  • Cho, Yong-Geun (Department of Mathematics Pohang University of Science and Technology) ;
  • Ozawa, Tohru (Department of Mathematics Hokkaido University)
  • Published : 2007.09.30
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Abstract

We consider initial value problems for the semirelativistic Hartree type equations with cubic convolution nonlinearity $F(u)=(V*{\mid}u{\mid}^2)u$. Here V is a sum of two Coulomb type potentials. Under a specified decay condition and a symmetric condition for the potential V we show the global existence and scattering of solutions.

Keywords

References

  1. 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
  2. Y. Cho and T. Ozawa, On radial solutions of semi-relativistic Hartree equations, to appear in Discrete and Continuous Dynamical System Series S
  3. Y. Cho, T. Ozawa, H. Sasaki, and Y. Shim, Remarks on the relativistic Hartree equations, Hokkaido Univ. Preprint Series in Math. #827 (2007)
  4. A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007), no. 4, 500-545 https://doi.org/10.1002/cpa.20134
  5. J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, Semin. Equ. Deriv. Partielles, Ecole Poly tech., Palaiseau, 2004
  6. J. Frohlich and E. Lenzmann, Blow-up for nonlinear wave equations describing Boson stars, to appear in Comm. Pure Appl. Math., arXiv:math-ph/0511003
  7. E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comrn. Math. Phys. 112 (1987), no. 1, 147-174 https://doi.org/10.1007/BF01217684
  8. E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, to appear in Mathematical Physics, Analysis and Geometry; arXiv:math.AP /0505456
  9. 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
  10. S. Machihara, K. Nakanishi, and T. Ozawa, Small global solutions and the nonrelaiiuistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoamericana 19 (2003), no. 1, 179-194
  11. T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schriulinqer equations, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 403-408 https://doi.org/10.1007/s00526-005-0349-2

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  1. Small data scattering for semi-relativistic equations with Hartree type nonlinearity vol.259, pp.10, 2015, https://doi.org/10.1016/j.jde.2015.06.037
  2. The Boson star equation with initial data of low regularity vol.97, 2014, https://doi.org/10.1016/j.na.2013.11.023
  3. Modified Scattering for the Boson Star Equation vol.332, pp.3, 2014, https://doi.org/10.1007/s00220-014-2094-x