Remark on a semirelativistic equation in the energy space

Previous Article
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$
PROC Home
This Issue
Next Article
Estimates for solutions of nonautonomous semilinear ill-posed problems

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

Abstract
Related Papers

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

[1]	Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[2]	Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147
[3]	Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[4]	Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
[5]	Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307
[6]	Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[7]	Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[8]	Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[9]	Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[10]	A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[11]	Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[12]	Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[13]	Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[14]	Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033
[15]	Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[16]	Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[17]	Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[18]	Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[19]	Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
[20]	Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149

Impact Factor:

American Institute of Mathematical Sciences