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

Global well-posedness and scattering for Skyrme wave maps

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

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.
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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