Large data global regularity for the classical equivariant Skyrme model

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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 and 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.
[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.
[3]	J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.
[4]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.
[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.
[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.
[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.
[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.
[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.
[10]	L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157. doi: 10.1080/03605302.2013.822885.
[11]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.
[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.
[13]	D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977.
[14]	D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780.
[15]	N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.
[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.
[17]	B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.
[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.
[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.
[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.
[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.
[22]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[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.
[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.

show all references

References:

[1]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
[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.
[3]	J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.
[4]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.
[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.
[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.
[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.
[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.
[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.
[10]	L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157. doi: 10.1080/03605302.2013.822885.
[11]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.
[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.
[13]	D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977.
[14]	D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780.
[15]	N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.
[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.
[17]	B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.
[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.
[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.
[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.
[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.
[22]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[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.
[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.

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