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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

D.-A. Geba and S. G. Rajeev, A continuity argument for a semilinear Skyrme model, Electron. J. Differential Equations, 2010 (2010), 1-9.  Google Scholar

[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.  Google Scholar

[5]

D.-A. Geba and S. G. Rajeev, Energy arguments for the Skyrme model,, work in progress., ().   Google Scholar

[6]

D.-A. Geba and D. da Silva, On the regularity of the $2+1$ dimensional Skyrme model,, preprint, ().   Google Scholar

[7]

M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Nuovo Cimento, 16 (1960), 705-726. doi: 10.1007/BF02859738.  Google Scholar

[8]

F. Gürsey, On the symmetries of strong and weak interactions, Nuovo Cimento, 16 (1960), 230-240. doi: 10.1007/BF02860276.  Google Scholar

[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.  Google Scholar

[10]

D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, preprint, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D.-A. Geba and S. G. Rajeev, A continuity argument for a semilinear Skyrme model, Electron. J. Differential Equations, 2010 (2010), 1-9.  Google Scholar

[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.  Google Scholar

[5]

D.-A. Geba and S. G. Rajeev, Energy arguments for the Skyrme model,, work in progress., ().   Google Scholar

[6]

D.-A. Geba and D. da Silva, On the regularity of the $2+1$ dimensional Skyrme model,, preprint, ().   Google Scholar

[7]

M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Nuovo Cimento, 16 (1960), 705-726. doi: 10.1007/BF02859738.  Google Scholar

[8]

F. Gürsey, On the symmetries of strong and weak interactions, Nuovo Cimento, 16 (1960), 230-240. doi: 10.1007/BF02860276.  Google Scholar

[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.  Google Scholar

[10]

D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, preprint, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.  Google Scholar

[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.  Google Scholar

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