American Institute of Mathematical Sciences

Advanced
November  2018, 38(11): 5537-5576. doi: 10.3934/dcds.2018244

Large data global regularity for the classical equivariant Skyrme model

Dan-Andrei Geba 1,, and  Manoussos G. Grillakis 2,

1. 

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA
2. 

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

* Corresponding author: Dan-Andrei Geba

Received  November 2017 Revised  May 2018 Published  August 2018

This article is concerned with the large data global regularity for the equivariant case of the classical Skyrme model and proves that this is valid for initial data in $H^s \times H^{s-1}(\mathbb{R}^3)$ with $s>7/2$.

Keywords: Skyrme model, global solutions, continuation criterion.
Mathematics Subject Classification: Primary: 35L70; Secondary: 81T13.
Citation: Dan-Andrei Geba, Manoussos G. Grillakis. Large data global regularity for the classical equivariant Skyrme model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5537-5576. doi: 10.3934/dcds.2018244
References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar

[2]

P. Bizoń, T. Chmaj and A. Rostworowski, Asymptotic stability of the skyrmion, Phys. Rev. D, 75 (2007), 121702, 5 pp. doi: 10.1103/PhysRevD.75.121702.  Google Scholar

[3]

J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.   Google Scholar

[4]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[5]

M. Creek, Large-Data Global Well-Posedness for the (1+2)-Dimensional Equivariant Faddeev Model, Thesis (Ph. D.)–University of Rochester. 2014, arXiv: 1310.4708.  Google Scholar

[6]

M. Creek, R. Donninger, W. Schlag and S. Snelson, Linear stability of the Skyrmion, Int. Math. Res. Not. IMRN, (2017), 2497-2537.  doi: 10.1093/imrn/rnw114.  Google Scholar

[7]

D.-A. Geba and M. G. Grillakis, An Introduction to the Theory of Wave Maps and Related Geometric Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.  Google Scholar

[8]

D.-A. Geba, K. Nakanishi and S. G. Rajeev, Global well-posedness and scattering for Skyrme wave maps, Commun. Pure Appl. Anal., 11 (2012), 1923-1933.  doi: 10.3934/cpaa.2012.11.1923.  Google Scholar

[9]

M. Gell-Mann and M. M. Lévy, The axial vector current in beta decay, Nuovo Cimento (10), 16 (1960), 705-726.  doi: 10.1007/BF02859738.  Google Scholar

[10]

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

[11]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.  Google Scholar

[12]

L. V. Kapitanskiǐ and O. A. Ladyzhenskaya, The Coleman principle for finding stationary points of invariant functionals, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 127 (1983), 84-102.   Google Scholar

[13]

D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977. Google Scholar

[14]

D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780. Google Scholar

[15]

N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.  Google Scholar

[16]

J. B. McLeod and W. C. Troy, The Skyrme model for nucleons under spherical symmetry, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 271-288.  doi: 10.1017/S0308210500029085.  Google Scholar

[17]

B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.  Google Scholar

[18]

J. Shatah, Weak solutions and development of singularities of the $\rm SU(2)$ $σ$-model, Comm. Pure Appl. Math., 41 (1988), 459-469.  doi: 10.1002/cpa.3160410405.  Google Scholar

[19]

T. H. R. Skyrme, A non-linear field theory, Proc. Roy. Soc. London Ser. A, 260 (1961), 127-138.  doi: 10.1098/rspa.1961.0018.  Google Scholar

[20]

T. H. R. Skyrme, Particle states of a quantized meson field, Proc. Roy. Soc. Ser. A, 262 (1961), 237-245.  doi: 10.1098/rspa.1961.0115.  Google Scholar

[21]

T. H. R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys., 31 (1962), 556-569.  doi: 10.1016/0029-5582(62)90775-7.  Google Scholar

[22]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[23]

N. Turok and D. Spergel, Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739.  doi: 10.1103/PhysRevLett.64.2736.  Google Scholar

[24]

W. W.-Y. Wong, Regular hyperbolicity, dominant energy condition and causality for Lagrangian theories of maps, Classical Quantum Gravity, 28 (2011), 215008, 23pp. doi: 10.1088/0264-9381/28/21/215008.  Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar

[2]

P. Bizoń, T. Chmaj and A. Rostworowski, Asymptotic stability of the skyrmion, Phys. Rev. D, 75 (2007), 121702, 5 pp. doi: 10.1103/PhysRevD.75.121702.  Google Scholar

[3]

J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.   Google Scholar

[4]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[5]

M. Creek, Large-Data Global Well-Posedness for the (1+2)-Dimensional Equivariant Faddeev Model, Thesis (Ph. D.)–University of Rochester. 2014, arXiv: 1310.4708.  Google Scholar

[6]

M. Creek, R. Donninger, W. Schlag and S. Snelson, Linear stability of the Skyrmion, Int. Math. Res. Not. IMRN, (2017), 2497-2537.  doi: 10.1093/imrn/rnw114.  Google Scholar

[7]

D.-A. Geba and M. G. Grillakis, An Introduction to the Theory of Wave Maps and Related Geometric Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.  Google Scholar

[8]

D.-A. Geba, K. Nakanishi and S. G. Rajeev, Global well-posedness and scattering for Skyrme wave maps, Commun. Pure Appl. Anal., 11 (2012), 1923-1933.  doi: 10.3934/cpaa.2012.11.1923.  Google Scholar

[9]

M. Gell-Mann and M. M. Lévy, The axial vector current in beta decay, Nuovo Cimento (10), 16 (1960), 705-726.  doi: 10.1007/BF02859738.  Google Scholar

[10]

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

[11]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.  Google Scholar

[12]

L. V. Kapitanskiǐ and O. A. Ladyzhenskaya, The Coleman principle for finding stationary points of invariant functionals, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 127 (1983), 84-102.   Google Scholar

[13]

D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977. Google Scholar

[14]

D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780. Google Scholar

[15]

N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.  Google Scholar

[16]

J. B. McLeod and W. C. Troy, The Skyrme model for nucleons under spherical symmetry, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 271-288.  doi: 10.1017/S0308210500029085.  Google Scholar

[17]

B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.  Google Scholar

[18]

J. Shatah, Weak solutions and development of singularities of the $\rm SU(2)$ $σ$-model, Comm. Pure Appl. Math., 41 (1988), 459-469.  doi: 10.1002/cpa.3160410405.  Google Scholar

[19]

T. H. R. Skyrme, A non-linear field theory, Proc. Roy. Soc. London Ser. A, 260 (1961), 127-138.  doi: 10.1098/rspa.1961.0018.  Google Scholar

[20]

T. H. R. Skyrme, Particle states of a quantized meson field, Proc. Roy. Soc. Ser. A, 262 (1961), 237-245.  doi: 10.1098/rspa.1961.0115.  Google Scholar

[21]

T. H. R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys., 31 (1962), 556-569.  doi: 10.1016/0029-5582(62)90775-7.  Google Scholar

[22]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[23]

N. Turok and D. Spergel, Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739.  doi: 10.1103/PhysRevLett.64.2736.  Google Scholar

[24]

W. W.-Y. Wong, Regular hyperbolicity, dominant energy condition and causality for Lagrangian theories of maps, Classical Quantum Gravity, 28 (2011), 215008, 23pp. doi: 10.1088/0264-9381/28/21/215008.  Google Scholar

[1]

Ruifeng Zhang, Nan Liu, Man An. Analytical solutions of Skyrme model. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2201-2211. doi: 10.3934/dcdss.2016092
[2]

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
[3]

Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
[4]

Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937
[5]

Stephen Schecter, Bradley J. Plohr, Dan Marchesin. Computation of Riemann solutions using the Dafermos regularization and continuation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 965-986. doi: 10.3934/dcds.2004.10.965
[6]

Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic & Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001
[7]

André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359
[8]

Liuyang Yuan, Zhongping Wan, Qiuhua Tang. A criterion for an approximation global optimal solution based on the filled functions. Journal of Industrial & Management Optimization, 2016, 12 (1) : 375-387. doi: 10.3934/jimo.2016.12.375
[9]

Konstantina Trivisa. Global existence and asymptotic analysis of solutions to a model for the dynamic combustion of compressible fluids. Conference Publications, 2003, 2003 (Special) : 852-863. doi: 10.3934/proc.2003.2003.852
[10]

Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[11]

Jinkai Li, Edriss Titi. Global well-posedness of strong solutions to a tropical climate model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4495-4516. doi: 10.3934/dcds.2016.36.4495
[12]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[13]

Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517
[14]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[15]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[16]

Josef Diblík, Klara Janglajew, Mária Kúdelčíková. An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2461-2467. doi: 10.3934/dcdsb.2014.19.2461
[17]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020110
[18]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613
[19]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043
[20]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (70)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences