\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Remark on a semirelativistic equation in the energy space

Abstract / Introduction Related Papers Cited by
  • Well-posedness of the Cauchy problem for a semirelativistic equation with cubic nonlinearity is shown in the energy space $H^{1/2}$. Solutions are constructed as a limit of approximation solutions, where the argument on the convergence depends on the completeness of $L^2$ and is independent of compactness. The Yudovitch type argument plays an important role for the convergence arguments.
    Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q55.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19.

    [2]

    Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.

    [3]

    J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.

    [4]

    K. Fujiwara, S. Machihara and T. Ozawa, On a system of semirelativistic equations in the energy space, Commun. Pure Appl. Anal., 14(2015), 1343-1355.

    [5]

    J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.

    [6]

    V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.

    [7]

    J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.

    [8]

    E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.

    [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), 982-988.

    [10]

    M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces, Publ. Res. Inst. Math. Sci., 37 (2001), 255-293.

    [11]

    T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769.

    [12]

    T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl., 155 (1991), 531-540.

    [13]

    T. Ozawa and N. Visciglia, An improvement on the brezis-gallout technique for 2d nls and 1d half-wave equation, arXiv:1403.7443.

    [14]

    I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135.

    [15]

    M. V. Vladimirov, On the solvability of a mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783.

  • 加载中
Open Access Under a Creative Commons license
SHARE

Article Metrics

HTML views() PDF downloads(148) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return