# American Institute of Mathematical Sciences

2015, 2015(special): 473-478. doi: 10.3934/proc.2015.0473

## Remark on a semirelativistic equation in the energy space

 1 Department of Pure and Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555 2 Faculty of Science, Saitama University, 255 Shimo-Okubo, Saitama 338-8570 3 Department of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

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.
Citation: Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. Remark on a semirelativistic equation in the energy space. Conference Publications, 2015, 2015 (special) : 473-478. doi: 10.3934/proc.2015.0473
##### References:
 [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,, , (). [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.

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##### References:
 [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,, , (). [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.
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