Global well-posedness and scattering for Skyrme wave maps

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September 2012, 11(5): 1923-1933. doi: 10.3934/cpaa.2012.11.1923

Global well-posedness and scattering for Skyrme wave maps

Dan-Andrei Geba ^1, , Kenji Nakanishi ^2, and Sarada G. Rajeev ^3,

1.	Department of Mathematics, University of Rochester, Rochester, NY 14627, United States
2.	Department of Mathematics, Kyoto University, Kyoto 606-8502
3.	Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, NY 14627, United States

Received June 2011 Revised September 2011 Published March 2012

Abstract
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We study equivariant solutions for two models ([13]-[15], [1]) arising in high energy physics, which are generalizations of the wave maps theory (i.e., the classical nonlinear $\sigma$ model) in 3 + 1 dimensions. We prove global existence and scattering for small initial data in critical Sobolev-Besov spaces.

Keywords: scattering theory., global existence, Skyrme model, Adkins-Nappi model, Wave maps.

Mathematics Subject Classification: Primary: 35L70, 81T1.

Citation: Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923

References:

[1]	G. Adkins and C. Nappi, Stabilization of chiral solitons via vector mesons, Phys. Lett. B, 137 (1984), 251-256. doi: 10.1016/0370-2693(84)90239-9.
[2]	P. Bizoń, T. Chmaj and A. Rostworowski, Asymptotic stability of the skyrmion, Phys. Rev. D, 75 (2007), 121702-121706. doi: 10.1103/PhysRevD.75.121702.
[3]	D.-A. Geba and S. G. Rajeev, A continuity argument for a semilinear Skyrme model, Electron. J. Differential Equations, 2010 (2010), 1-9.
[4]	D.-A. Geba and S. G. Rajeev, Nonconcentration of energy for a semilinear Skyrme model, Ann. Physics, 325 (2010), 2697-2706. doi: 10.1016/j.aop.2010.07.002.
[5]	D.-A. Geba and S. G. Rajeev, Energy arguments for the Skyrme model, work in progress.
[6]	D.-A. Geba and D. da Silva, On the regularity of the $2+1$ dimensional Skyrme model, preprint, arXiv:1106.3974.
[7]	M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Nuovo Cimento, 16 (1960), 705-726. doi: 10.1007/BF02859738.
[8]	F. Gürsey, On the symmetries of strong and weak interactions, Nuovo Cimento, 16 (1960), 230-240. doi: 10.1007/BF02860276.
[9]	F. Gürsey, On the structure and parity of weak interaction currents, Ann. Physics, 12 (1961), 91-117. doi: 10.1016/0003-4916(61)90147-6.
[10]	D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, preprint, 2011.
[11]	F. Lin and Y. Yang, Analysis on Faddeev knots and Skyrme solitons: recent progress and open problems, Perspectives in nonlinear partial differential equations, Contemp. Math., 446 (2007), 319-344. doi: 10.1090/conm/446/08639.
[12]	J. Shatah, Weak solutions and development of singularities of the SU(2) $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469. doi: 10.1002/cpa.3160410405.
[13]	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.
[14]	T. H. R. Skyrme, Particle states of a quantized meson field, Proc. Roy. Soc. London Ser. A, 262 (1961), 237-245. doi: 10.1098/rspa.1961.0115.
[15]	T. H. R. Skyrme, A unified field theory of mesons and baryons, Nuclear Phys., 31 (1962), 556-569. doi: 10.1016/0029-5582(62)90775-7.
[16]	D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.
[17]	N. Turok and D. Spergel, Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739. doi: 10.1103/PhysRevLett.64.2736.

show all references

References:

[1]	G. Adkins and C. Nappi, Stabilization of chiral solitons via vector mesons, Phys. Lett. B, 137 (1984), 251-256. doi: 10.1016/0370-2693(84)90239-9.
[2]	P. Bizoń, T. Chmaj and A. Rostworowski, Asymptotic stability of the skyrmion, Phys. Rev. D, 75 (2007), 121702-121706. doi: 10.1103/PhysRevD.75.121702.
[3]	D.-A. Geba and S. G. Rajeev, A continuity argument for a semilinear Skyrme model, Electron. J. Differential Equations, 2010 (2010), 1-9.
[4]	D.-A. Geba and S. G. Rajeev, Nonconcentration of energy for a semilinear Skyrme model, Ann. Physics, 325 (2010), 2697-2706. doi: 10.1016/j.aop.2010.07.002.
[5]	D.-A. Geba and S. G. Rajeev, Energy arguments for the Skyrme model, work in progress.
[6]	D.-A. Geba and D. da Silva, On the regularity of the $2+1$ dimensional Skyrme model, preprint, arXiv:1106.3974.
[7]	M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Nuovo Cimento, 16 (1960), 705-726. doi: 10.1007/BF02859738.
[8]	F. Gürsey, On the symmetries of strong and weak interactions, Nuovo Cimento, 16 (1960), 230-240. doi: 10.1007/BF02860276.
[9]	F. Gürsey, On the structure and parity of weak interaction currents, Ann. Physics, 12 (1961), 91-117. doi: 10.1016/0003-4916(61)90147-6.
[10]	D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, preprint, 2011.
[11]	F. Lin and Y. Yang, Analysis on Faddeev knots and Skyrme solitons: recent progress and open problems, Perspectives in nonlinear partial differential equations, Contemp. Math., 446 (2007), 319-344. doi: 10.1090/conm/446/08639.
[12]	J. Shatah, Weak solutions and development of singularities of the SU(2) $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469. doi: 10.1002/cpa.3160410405.
[13]	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.
[14]	T. H. R. Skyrme, Particle states of a quantized meson field, Proc. Roy. Soc. London Ser. A, 262 (1961), 237-245. doi: 10.1098/rspa.1961.0115.
[15]	T. H. R. Skyrme, A unified field theory of mesons and baryons, Nuclear Phys., 31 (1962), 556-569. doi: 10.1016/0029-5582(62)90775-7.
[16]	D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.
[17]	N. Turok and D. Spergel, Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739. doi: 10.1103/PhysRevLett.64.2736.

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