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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, 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

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