JournalsaihpcVol. 26, No. 5pp. 1831–1852

Global well-posedness and scattering for the defocusing H12 H^{\frac{1}{2}} -subcritical Hartree equation in Rd R^{d}

  • Changxing Miao

    Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
  • Guixiang Xu

    Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
  • Lifeng Zhao

    Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
Global well-posedness and scattering for the defocusing \( H^{\frac{1}{2}} \)-subcritical Hartree equation in \( R^{d} \) cover
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Abstract

We prove the global well-posedness and scattering for the defocusing H12H^{\frac{1}{2}}-subcritical (that is, 2<γ<32 < \gamma < 3) Hartree equation with low regularity data in Rd\mathbb{R}^{d}, d3d⩾3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space Hs(Rd)H^{s}(\mathbb{R}^{d}) with s>4(γ2)/(3γ4)s > 4(\gamma −2)/ (3\gamma −4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s>max(1/2,4(γ2)/(3γ4))s > \mathrm{\max }(1/ 2,4(\gamma −2)/ (3\gamma −4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ)ξpm(\xi ) \cdot 〈\xi 〉^{p}. As a byproduct of our proof, we obtain that the HsH^{s} norm of the solution obeys the uniform-in-time bounds.

Cite this article

Changxing Miao, Guixiang Xu, Lifeng Zhao, Global well-posedness and scattering for the defocusing H12 H^{\frac{1}{2}} -subcritical Hartree equation in Rd R^{d} . Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1831–1852

DOI 10.1016/J.ANIHPC.2009.01.003