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

Previous Article
Uniform hyperbolicity in nonflat billiards
DCDS Home
This Issue
Next Article
Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

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 and Continuous Dynamical Systems, 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.
[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.

[1]	Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[2]	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 and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[3]	Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure and Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
[4]	Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[5]	Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
[6]	Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[7]	Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
[8]	Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
[9]	Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[10]	Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
[11]	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
[12]	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 and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[13]	Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034
[14]	Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[15]	Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923
[16]	Adán J. Corcho. Ill-Posedness for the Benney system. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965
[17]	Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $\|u\|^2$. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395
[18]	In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012
[19]	Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations and Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002
[20]	Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Van Duong Dinh
on Google Scholar
Van Duong Dinh

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Van Duong Dinh
on Google Scholar
Van Duong Dinh

[Back to Top]