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July  2015, 14(4): 1343-1355. doi: 10.3934/cpaa.2015.14.1343

On a system of semirelativistic equations in the energy space

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

Received  June 2014 Revised  June 2014 Published  April 2015

Well-posedness of the Cauchy problem for a system of semirelativistic equations is shown in the energy space. Solutions are constructed as a limit of an approximate solutions. A Yudovitch type argument plays an important role for the convergence arguments.
Citation: Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. On a system of semirelativistic equations in the energy space. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1343-1355. doi: 10.3934/cpaa.2015.14.1343
References:
 [1] J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4726198. Google Scholar [2] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, \emph{Comm. Partial Differential Equations}, 5 (1980), 773. doi: 10.1080/03605308008820154. Google Scholar [3] R. Carles and T. Ozawa, Finite time extinction for nonlinear Schrödinger equation in 1D and 2D,, \emph{Comm. Partial Differential Equation}, 40 (2015), 897. doi: 10.1080/03605302.2014.967356. Google Scholar [4] T. Cazenave, Semilinear Schrödinger Equations,, American Mathematical Society, (2003). Google Scholar [5] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060. doi: 10.1137/060653688. Google Scholar [6] J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1691. doi: 10.1002/cpa.20186. Google Scholar [7] K. Fujiwara, S. Machihara and T. Ozawa, Well-posedness for the Cauchy problem of a system of semirelativistic equations,, \emph{Commun. Math. Phys.}, (). Google Scholar [8] N. Hayashi, C. Li and P. I. Naumkin, On a system of nonlinear Schrödinger equations in 2D,, \emph{Differential Integral Equations}, 24 (2011), 417. Google Scholar [9] N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations,, \emph{Differ. Equ. Appl.}, 3 (2011), 415. doi: 10.7153/dea-03-26. Google Scholar [10] N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 661. doi: 10.1016/j.anihpc.2012.10.007. Google Scholar [11] N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations,, \emph{J. Math. Soc. Japan}, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar [12] G. Hoshino and T. Ozawa, Analytic smoothing effect for a system of nonlinear Schr\"odinger equations,, \emph{Differ. Equ. Appl.}, 5 (2013), 395. doi: 10.7153/dea-05-25. Google Scholar [13] V. I. Judovič, Non-stationary flows of an ideal incompressible fluid,, \emph{\u Z. Vy\v cisl. Mat. i Mat. Fiz.}, 3 (1963), 1032. Google Scholar [14] J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation,, \emph{Arch. Ration. Mech. Anal.}, 209 (2013), 61. doi: 10.1007/s00205-013-0620-1. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type,, \emph{Math. Phys. Anal. Geom.}, 10 (2007), 43. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, \emph{Nonlinear Anal.}, 14 (1990), 765. doi: 10.1016/0362-546X(90)90104-O. Google Scholar [18] T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,, \emph{J. Math. Anal. Appl.}, 155 (1991), 531. doi: 10.1016/0022-247X(91)90017-T. Google Scholar [19] T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,, \emph{Commun. Math. Phys.}, 245 (2004), 105. doi: 10.1007/s00220-003-1004-4. Google Scholar [20] T. Ozawa and N. Visciglia, An improvement on the Brezis-Gallouet technique for 2D NLS and 1D half-wave equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, (). Google Scholar [21] G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Differential Integral Equations}, 4 (1991), 527. Google Scholar [22] M. V. Vladimirov, On the solvability of a mixed problem for a nonlinear equation of Schrödinger type,, \emph{Dokl. Akad. Nauk SSSR}, 275 (1984), 780. Google Scholar

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References:
 [1] J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4726198. Google Scholar [2] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, \emph{Comm. Partial Differential Equations}, 5 (1980), 773. doi: 10.1080/03605308008820154. Google Scholar [3] R. Carles and T. Ozawa, Finite time extinction for nonlinear Schrödinger equation in 1D and 2D,, \emph{Comm. Partial Differential Equation}, 40 (2015), 897. doi: 10.1080/03605302.2014.967356. Google Scholar [4] T. Cazenave, Semilinear Schrödinger Equations,, American Mathematical Society, (2003). Google Scholar [5] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060. doi: 10.1137/060653688. Google Scholar [6] J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1691. doi: 10.1002/cpa.20186. Google Scholar [7] K. Fujiwara, S. Machihara and T. Ozawa, Well-posedness for the Cauchy problem of a system of semirelativistic equations,, \emph{Commun. Math. Phys.}, (). Google Scholar [8] N. Hayashi, C. Li and P. I. Naumkin, On a system of nonlinear Schrödinger equations in 2D,, \emph{Differential Integral Equations}, 24 (2011), 417. Google Scholar [9] N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations,, \emph{Differ. Equ. Appl.}, 3 (2011), 415. doi: 10.7153/dea-03-26. Google Scholar [10] N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 661. doi: 10.1016/j.anihpc.2012.10.007. Google Scholar [11] N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations,, \emph{J. Math. Soc. Japan}, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar [12] G. Hoshino and T. Ozawa, Analytic smoothing effect for a system of nonlinear Schr\"odinger equations,, \emph{Differ. Equ. Appl.}, 5 (2013), 395. doi: 10.7153/dea-05-25. Google Scholar [13] V. I. Judovič, Non-stationary flows of an ideal incompressible fluid,, \emph{\u Z. Vy\v cisl. Mat. i Mat. Fiz.}, 3 (1963), 1032. Google Scholar [14] J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation,, \emph{Arch. Ration. Mech. Anal.}, 209 (2013), 61. doi: 10.1007/s00205-013-0620-1. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type,, \emph{Math. Phys. Anal. Geom.}, 10 (2007), 43. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, \emph{Nonlinear Anal.}, 14 (1990), 765. doi: 10.1016/0362-546X(90)90104-O. Google Scholar [18] T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,, \emph{J. Math. Anal. Appl.}, 155 (1991), 531. doi: 10.1016/0022-247X(91)90017-T. Google Scholar [19] T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,, \emph{Commun. Math. Phys.}, 245 (2004), 105. doi: 10.1007/s00220-003-1004-4. Google Scholar [20] T. Ozawa and N. Visciglia, An improvement on the Brezis-Gallouet technique for 2D NLS and 1D half-wave equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, (). Google Scholar [21] G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Differential Integral Equations}, 4 (1991), 527. Google Scholar [22] M. V. Vladimirov, On the solvability of a mixed problem for a nonlinear equation of Schrödinger type,, \emph{Dokl. Akad. Nauk SSSR}, 275 (1984), 780. Google Scholar
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