American Institute of Mathematical Sciences

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

On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

Van Duong Dinh ,

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.

Keywords: Nonlinear half-wave equation, local well-posedness, Strichartz estimates, small data scattering, ill-posedness.
Mathematics Subject Classification: Primary: 35A01, 35E15; Secondary: 35Q55.
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. 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.  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. 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.  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. 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.  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. 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.  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

[1]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[2]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
[3]

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
[4]

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
[5]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
[6]

Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
[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]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[9]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[10]

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923
[11]

Adán J. Corcho. Ill-Posedness for the Benney system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965
[12]

Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671
[13]

Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395
[14]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012
[15]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[16]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[17]

Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002
[18]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
[19]

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
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (59)
  • HTML views (109)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences