March  2018, 38(3): 1127-1143. doi: 10.3934/dcds.2018047

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

* Corresponding author

Received  January 2017 Revised  October 2017 Published  December 2017

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 [6] to prove the ill-posedness for the (NLHW) in a certain range of the super-critical case.

Citation: Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047
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.  Google Scholar

[2] J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.   Google Scholar
[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003.  Google Scholar

[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.  Google Scholar

[6]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv:math/0311048. Google Scholar

[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.  Google Scholar

[8]

J. CollianderM. KeelG. StaffilaniH. 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.  Google Scholar

[9]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, preprint, arXiv:1609.06181. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[20]

J. KriegerE. 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.  Google Scholar

[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.  Google Scholar

[22]

G. Staffilani, The Initial Value Problem for Some Dispersive Differential Equations, Ph. D thesis, University of Chicago, 1995.  Google Scholar

[23]

M. Taylor, Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, 2000.  Google Scholar

[24]

H. Triebel, Theory of Function Spaces, Basel: Birkhäuser, 1983.  Google Scholar

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.  Google Scholar

[2] J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.   Google Scholar
[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003.  Google Scholar

[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.  Google Scholar

[6]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv:math/0311048. Google Scholar

[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.  Google Scholar

[8]

J. CollianderM. KeelG. StaffilaniH. 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.  Google Scholar

[9]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, preprint, arXiv:1609.06181. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[20]

J. KriegerE. 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.  Google Scholar

[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.  Google Scholar

[22]

G. Staffilani, The Initial Value Problem for Some Dispersive Differential Equations, Ph. D thesis, University of Chicago, 1995.  Google Scholar

[23]

M. Taylor, Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, 2000.  Google Scholar

[24]

H. Triebel, Theory of Function Spaces, Basel: Birkhäuser, 1983.  Google Scholar

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