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On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces
Institut de Mathématiques de Toulouse UMR5219, Université Toulouse CNRS, 31062 Toulouse Cedex 9, France |
We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs are mainly based on the contraction mapping argument using Strichartz estimates. We also apply the technique of Christ-Colliander-Tao in [
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Non-Linear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics 343, Springer-Verlag, Berlin, 2011. |
[2] |
J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.
![]() ![]() |
[3] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[4] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003. |
[5] |
A. Choffrut and O. Pocovnicu,
Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2016 (2016), 1-40.
doi: 10.1093/imrn/rnw246. |
[6] |
M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv:math/0311048. |
[7] |
M. Christ and I. Weinstein,
Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.
doi: 10.1016/0022-1236(91)90103-C. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[9] |
V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, preprint, arXiv:1609.06181. |
[10] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commum. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[11] |
J. Fröhlich and E. Lenzmann,
Blowup for nonlinear wave equations describing boson stars, Commum. Pure Appl. Math., 60 (2007), 1691-1705.
doi: 10.1002/cpa.20186. |
[12] |
K. Fujiwara, V. Georgiev and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, preprint, arXiv:1611.09674. |
[13] |
K. Fujiwara and T. Ozawa,
Remarks on global solutions to the Cauchy problem for semirelativistic equations with power type nonlinearity, Int. J. Math. Anal., 9 (2015), 2599-2610.
doi: 10.12988/ijma.2015.58211. |
[14] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
doi: 10.1007/BF01168155. |
[15] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[16] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[17] |
A. D. Ionescu and F. Pusateri,
Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.
doi: 10.1016/j.jfa.2013.08.027. |
[18] |
T. Kato,
On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[19] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[20] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Rational Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[21] |
O. Pocovnicu,
First and second order approximations for a nonlinear wave equation, J. Dynam. Differential Equations, 25 (2013), 305-333.
doi: 10.1007/s10884-013-9286-5. |
[22] |
G. Staffilani,
The Initial Value Problem for Some Dispersive Differential Equations, Ph. D thesis, University of Chicago, 1995. |
[23] |
M. Taylor,
Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, 2000. |
[24] |
H. Triebel,
Theory of Function Spaces, Basel: Birkhäuser, 1983. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Non-Linear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics 343, Springer-Verlag, Berlin, 2011. |
[2] |
J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.
![]() ![]() |
[3] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[4] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003. |
[5] |
A. Choffrut and O. Pocovnicu,
Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2016 (2016), 1-40.
doi: 10.1093/imrn/rnw246. |
[6] |
M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv:math/0311048. |
[7] |
M. Christ and I. Weinstein,
Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.
doi: 10.1016/0022-1236(91)90103-C. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[9] |
V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, preprint, arXiv:1609.06181. |
[10] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commum. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[11] |
J. Fröhlich and E. Lenzmann,
Blowup for nonlinear wave equations describing boson stars, Commum. Pure Appl. Math., 60 (2007), 1691-1705.
doi: 10.1002/cpa.20186. |
[12] |
K. Fujiwara, V. Georgiev and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, preprint, arXiv:1611.09674. |
[13] |
K. Fujiwara and T. Ozawa,
Remarks on global solutions to the Cauchy problem for semirelativistic equations with power type nonlinearity, Int. J. Math. Anal., 9 (2015), 2599-2610.
doi: 10.12988/ijma.2015.58211. |
[14] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
doi: 10.1007/BF01168155. |
[15] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[16] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[17] |
A. D. Ionescu and F. Pusateri,
Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.
doi: 10.1016/j.jfa.2013.08.027. |
[18] |
T. Kato,
On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[19] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[20] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Rational Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[21] |
O. Pocovnicu,
First and second order approximations for a nonlinear wave equation, J. Dynam. Differential Equations, 25 (2013), 305-333.
doi: 10.1007/s10884-013-9286-5. |
[22] |
G. Staffilani,
The Initial Value Problem for Some Dispersive Differential Equations, Ph. D thesis, University of Chicago, 1995. |
[23] |
M. Taylor,
Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, 2000. |
[24] |
H. Triebel,
Theory of Function Spaces, Basel: Birkhäuser, 1983. |
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