Large data global regularity for the classical equivariant Skyrme model

Dan-Andrei Geba; Manoussos G. Grillakis

doi:10.3934/dcds.2018244

Article Contents

2018, Volume 38, Issue 11: 5537-5576. Doi: 10.3934/dcds.2018244

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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
* Corresponding author: Dan-Andrei Geba

Received: October 31, 2017

Revised: April 30, 2018

Published: August 2018

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Abstract

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 $s>7/2$.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 81T13.

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References

	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
	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.
	J. Bourgain and D. Li , On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014) , 1037-1072.
	Y. Cho and T. Ozawa , Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009) , 355-365. doi: 10.1142/S0219199709003399.
	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.
	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.
	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.
	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.
	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.
	L. Grafakos and S. Oh , The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014) , 1128-1157. doi: 10.1080/03605302.2013.822885.
	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.
	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.
	D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977.
	D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780.
	N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.
	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.
	B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.
	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.
	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.
	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.
	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.
	W. A. Strauss , Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977) , 149-162. doi: 10.1007/BF01626517.
	N. Turok and D. Spergel , Global texture and the microwave background, Phys. Rev. Lett., 64 (1990) , 2736-2739. doi: 10.1103/PhysRevLett.64.2736.
	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.