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Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces
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, J. Math. Phys., 53 (2012), 062301, 19. doi: 10.1063/1.4726198.  Google Scholar [2] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar [3] R. Carles and T. Ozawa, Finite time extinction for nonlinear Schrödinger equation in 1D and 2D, Comm. Partial Differential Equation, 40 (2015), 897-917. 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, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.  Google Scholar [6] J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705. 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, Differential Integral Equations, 24 (2011), 417-434.  Google Scholar [9] N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. 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, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.  Google Scholar [12] G. Hoshino and T. Ozawa, Analytic smoothing effect for a system of nonlinear Schr\"odinger equations, Differ. Equ. Appl., 5 (2013), 395-408. doi: 10.7153/dea-05-25.  Google Scholar [13] V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.  Google Scholar [14] 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. doi: 10.1007/s00205-013-0620-1.  Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64. 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, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. 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, J. Math. Anal. Appl., 155 (1991), 531-540. 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, Commun. Math. Phys., 245 (2004), 105-121. 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, Differential Integral Equations, 4 (1991), 527-542.  Google Scholar [22] 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.  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), 062301, 19. doi: 10.1063/1.4726198.  Google Scholar [2] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar [3] R. Carles and T. Ozawa, Finite time extinction for nonlinear Schrödinger equation in 1D and 2D, Comm. Partial Differential Equation, 40 (2015), 897-917. 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, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.  Google Scholar [6] J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705. 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, Differential Integral Equations, 24 (2011), 417-434.  Google Scholar [9] N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. 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, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.  Google Scholar [12] G. Hoshino and T. Ozawa, Analytic smoothing effect for a system of nonlinear Schr\"odinger equations, Differ. Equ. Appl., 5 (2013), 395-408. doi: 10.7153/dea-05-25.  Google Scholar [13] V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.  Google Scholar [14] 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. doi: 10.1007/s00205-013-0620-1.  Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64. 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, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. 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, J. Math. Anal. Appl., 155 (1991), 531-540. 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, Commun. Math. Phys., 245 (2004), 105-121. 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, Differential Integral Equations, 4 (1991), 527-542.  Google Scholar [22] 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.  Google Scholar
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