## On the regularity of the $2+1$ dimensional equivariant Skyrme model

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- by Dan-Andrei Geba and Daniel da Silva PDF
- Proc. Amer. Math. Soc.
**141**(2013), 2105-2115 Request permission

## Abstract:

One of the most interesting open problems concerning the Skyrme model of nuclear physics is the regularity of its solutions. In this article, we study $2+1$ dimensional equivariant Skyrme maps, for which we prove, using the method of multipliers, that the energy does not concentrate. This is one of the important steps towards a global regularity theory.## References

- Piotr Bizoń, Tadeusz Chmaj, and Andrzej Rostworowski,
*Asymptotic stability of the skyrmion*, Phys. Rev. D**75**(2007), no. 12, 121702, 5. MR**2326835**, DOI 10.1103/PhysRevD.75.121702 - Piero D’Ancona and Vladimir Georgiev,
*Wave maps and ill-posedness of their Cauchy problem*, New trends in the theory of hyperbolic equations, Oper. Theory Adv. Appl., vol. 159, Birkhäuser, Basel, 2005, pp. 1–111. MR**2175916**, DOI 10.1007/3-7643-7386-5_{1} - Dan-Andrei Geba, Kenji Nakanishi, and Sarada G. Rajeev,
*Global well-posedness and scattering for Skyrme wave maps*, Commun. Pure Appl. Anal.**11**(2012), no. 5, 1923–1933. MR**2911118**, DOI 10.3934/cpaa.2012.11.1923 - M. Gell-Mann and M. Lévy,
*The axial vector current in beta decay*, Nuovo Cimento (10)**16**(1960), 705–726 (English, with Italian summary). MR**140316**, DOI 10.1007/BF02859738 - F. Gürsey,
*On the symmetries of strong and weak interactions*, Nuovo Cimento (10)**16**(1960), 230–240 (English, with Italian summary). MR**118418**, DOI 10.1007/BF02860276 - —,
*On the structure and parity of weak interaction currents*, Ann. Physics**12**(1961), no. 1, 91–117. - Frédéric Hélein,
*Harmonic maps, conservation laws and moving frames*, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR**1913803**, DOI 10.1017/CBO9780511543036 - J. Krieger,
*Global regularity and singularity development for wave maps*, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 167–201. MR**2488946**, DOI 10.4310/SDG.2007.v12.n1.a5 - J. Krieger, W. Schlag, and D. Tataru,
*Renormalization and blow up for charge one equivariant critical wave maps*, Invent. Math.**171**(2008), no. 3, 543–615. MR**2372807**, DOI 10.1007/s00222-007-0089-3 - A. N. Leznov, B. Piette, and W. J. Zakrzewski,
*On the integrability of pure Skyrme models in two dimensions*, J. Math. Phys.**38**(1997), no. 6, 3007–3011. MR**1449544**, DOI 10.1063/1.532029 - D. Li,
*Global wellposedness of hedgehog solutions for the $(3+1)$ Skyrme model*, preprint, 2011. - Fanghua Lin and Yisong Yang,
*Existence of two-dimensional skyrmions via the concentration-compactness method*, Comm. Pure Appl. Math.**57**(2004), no. 10, 1332–1351. MR**2070206**, DOI 10.1002/cpa.20038 - Fanghua Lin and Yisong Yang,
*Analysis on Faddeev knots and Skyrme solitons: recent progress and open problems*, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 319–344. MR**2376667**, DOI 10.1090/conm/446/08639 - Pierre Raphaël and Igor Rodnianski,
*Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems*, Publ. Math. Inst. Hautes Études Sci.**115**(2012), 1–122. MR**2929728**, DOI 10.1007/s10240-011-0037-z - Igor Rodnianski and Jacob Sterbenz,
*On the formation of singularities in the critical $\textrm {O}(3)$ $\sigma$-model*, Ann. of Math. (2)**172**(2010), no. 1, 187–242. MR**2680419**, DOI 10.4007/annals.2010.172.187 - Jalal Shatah,
*Weak solutions and development of singularities of the $\textrm {SU}(2)$ $\sigma$-model*, Comm. Pure Appl. Math.**41**(1988), no. 4, 459–469. MR**933231**, DOI 10.1002/cpa.3160410405 - J. Shatah and A. Tahvildar-Zadeh,
*Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds*, Comm. Pure Appl. Math.**45**(1992), no. 8, 947–971. MR**1168115**, DOI 10.1002/cpa.3160450803 - T. H. R. Skyrme,
*A non-linear field theory*, Proc. Roy. Soc. London Ser. A**260**(1961), 127–138. MR**128862**, DOI 10.1098/rspa.1961.0018 - T. H. R. Skyrme,
*Particle states of a quantized meson field*, Proc. Roy. Soc. London Ser. A**262**(1961), 237–245. MR**153323** - T. H. R. Skyrme,
*A unified field theory of mesons and baryons*, Nuclear Phys.**31**(1962), 556–569. MR**0138394**, DOI 10.1016/0029-5582(62)90775-7 - Daniel Tataru,
*The wave maps equation*, Bull. Amer. Math. Soc. (N.S.)**41**(2004), no. 2, 185–204. MR**2043751**, DOI 10.1090/S0273-0979-04-01005-5

## Additional Information

**Dan-Andrei Geba**- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
**Daniel da Silva**- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- Received by editor(s): October 6, 2011
- Published electronically: February 7, 2013
- Communicated by: James E. Colliander
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 2105-2115 - MSC (2010): Primary 35L70, 81T13
- DOI: https://doi.org/10.1090/S0002-9939-2013-11865-4
- MathSciNet review: 3034436