• Previous Article
    Modified wave operators without loss of regularity for some long range Hartree equations. II
  • CPAA Home
  • This Issue
  • Next Article
    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,, \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

show all references

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

[1]

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

[2]

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

[3]

Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Didier Pilod. Sharp well-posedness results for the Kuramoto-Velarde equation. Communications on Pure & Applied Analysis, 2008, 7 (4) : 867-881. doi: 10.3934/cpaa.2008.7.867

[19]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[20]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]