July  2022, 42(7): 3103-3118. doi: 10.3934/dcds.2022010

Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation

Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea

*Corresponding author: Sangdon Jin

Received  November 2021 Published  July 2022 Early access  February 2022

For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [2]. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.

Citation: Younghun Hong, Sangdon Jin. Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3103-3118. doi: 10.3934/dcds.2022010
References:
[1]

J. BellazziniN. BoussaïdL. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.

[2]

J. BellazziniV. Georgiev and N. Visciglia, Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.  doi: 10.1007/s00208-018-1666-z.

[3]

J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp. doi: 10.1063/1.4726198.

[4]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[7]

W. ChoiY. Hong and J. Seok, On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1241-1263.  doi: 10.1017/prm.2018.114.

[8]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[9]

J. FröhlichB. L. G. Jonsson and E. Lenzmann, Effective dynamics for boson stars, Nonlinearity, 20 (2007), 1031-1075.  doi: 10.1088/0951-7715/20/5/001.

[10]

K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 473–478. doi: 10.3934/proc.2015.0473.

[11]

J. KriegerE. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.

[12]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[13]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[14]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[15]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/s0294-1449(16)30422-x.

[17]

X. Luo and T. Yang, Stable solitary waves for pseudo-relativistic Hartree equations with short range potential, Nonlinear Anal., 207 (2021), Paper No. 112275, 13 pp. doi: 10.1016/j.na.2021.112275.

[18]

J. Seok and Y. Hong, Ground states to the generalized nonlinear schrödinger equations with bernstein symbols, Anal. Theory Appl., 37 (2021), 157-177.  doi: 10.4208/ata.2021.pr80.06.

[19]

Q. Shi and C. Peng, Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[20]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

show all references

References:
[1]

J. BellazziniN. BoussaïdL. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.

[2]

J. BellazziniV. Georgiev and N. Visciglia, Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.  doi: 10.1007/s00208-018-1666-z.

[3]

J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp. doi: 10.1063/1.4726198.

[4]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[7]

W. ChoiY. Hong and J. Seok, On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1241-1263.  doi: 10.1017/prm.2018.114.

[8]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[9]

J. FröhlichB. L. G. Jonsson and E. Lenzmann, Effective dynamics for boson stars, Nonlinearity, 20 (2007), 1031-1075.  doi: 10.1088/0951-7715/20/5/001.

[10]

K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 473–478. doi: 10.3934/proc.2015.0473.

[11]

J. KriegerE. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.

[12]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[13]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[14]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[15]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/s0294-1449(16)30422-x.

[17]

X. Luo and T. Yang, Stable solitary waves for pseudo-relativistic Hartree equations with short range potential, Nonlinear Anal., 207 (2021), Paper No. 112275, 13 pp. doi: 10.1016/j.na.2021.112275.

[18]

J. Seok and Y. Hong, Ground states to the generalized nonlinear schrödinger equations with bernstein symbols, Anal. Theory Appl., 37 (2021), 157-177.  doi: 10.4208/ata.2021.pr80.06.

[19]

Q. Shi and C. Peng, Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[20]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

[1]

Xueying Chen, Guanfeng Li, Sijia Bao. Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1755-1772. doi: 10.3934/cpaa.2022045

[2]

Yuxia Guo, Shaolong Peng. Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1637-1648. doi: 10.3934/cpaa.2022037

[3]

Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic and Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002

[4]

Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure and Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963

[5]

Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure and Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365

[6]

Xuecheng Wang. Decay estimates for the $ 3D $ relativistic and non-relativistic Vlasov-Poisson systems. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022019

[7]

Rowan Killip, Soonsik Kwon, Shuanglin Shao, Monica Visan. On the mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 191-221. doi: 10.3934/dcds.2012.32.191

[8]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156

[9]

Benjamin Dodson, Cristian Gavrus. Instability of the soliton for the focusing, mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1767-1799. doi: 10.3934/dcds.2021171

[10]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure and Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029

[11]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[12]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[13]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[14]

Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141

[15]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic and Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725

[16]

Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275

[17]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

[18]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic and Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[19]

Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic and Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014

[20]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2279-2290. doi: 10.3934/cpaa.2021069

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (228)
  • HTML views (126)
  • Cited by (0)

Other articles
by authors

[Back to Top]