American Institute of Mathematical Sciences

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

Remark on a semirelativistic equation in the energy space

Kazumasa Fujiwara 1, , Shuji Machihara 2, and  Tohru Ozawa 3,

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.
Keywords: cubic nonlinearity, Semirelativistic equation, compactness argument., well-posedness, Yudovitch argument.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q5.
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).   Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[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.   Google Scholar

[6]

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

[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.   Google Scholar

[8]

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

[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.   Google Scholar

[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.   Google Scholar

[11]

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

[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.   Google Scholar

[13]

T. Ozawa and N. Visciglia, An improvement on the brezis-gallout technique for 2d nls and 1d half-wave equation,, , ().   Google Scholar

[14]

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

[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.   Google Scholar

show all references

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).   Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[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.   Google Scholar

[6]

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

[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.   Google Scholar

[8]

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

[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.   Google Scholar

[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.   Google Scholar

[11]

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

[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.   Google Scholar

[13]

T. Ozawa and N. Visciglia, An improvement on the brezis-gallout technique for 2d nls and 1d half-wave equation,, , ().   Google Scholar

[14]

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

[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.   Google Scholar

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