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Global existence for a two-component Camassa-Holm system with an arbitrary smooth function
Large data global regularity for the classical equivariant Skyrme model
1. | Department of Mathematics, University of Rochester, Rochester, NY 14627, USA |
2. | Department of Mathematics, University of Maryland, College Park, MD 20742, USA |
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 $s>7/2$.
References:
[1] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. |
[2] |
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. |
[3] |
J. Bourgain and D. Li,
On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.
|
[4] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[11] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997. |
[12] |
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.
|
[13] |
D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977. |
[14] |
D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780. |
[15] |
N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617034. |
[16] |
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. |
[17] |
B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[23] |
N. Turok and D. Spergel,
Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739.
doi: 10.1103/PhysRevLett.64.2736. |
[24] |
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. |
show all references
References:
[1] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. |
[2] |
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. |
[3] |
J. Bourgain and D. Li,
On an endpoint Kato-Ponce inequality, Differential Integral Equations, 27 (2014), 1037-1072.
|
[4] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[11] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997. |
[12] |
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.
|
[13] |
D. Li, Global wellposedness of hedgehog solutions for the ($3+1$) Skyrme model, 2012, Preprint, arXiv: 1208.4977. |
[14] |
D. Li, On Kato-Ponce and fractional Leibniz, 2016, Preprint, arXiv: 1609.01780. |
[15] |
N. S. Manton and P. M. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617034. |
[16] |
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. |
[17] |
B. Opic and A. Kufner, Hardy-type Inequalities vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[23] |
N. Turok and D. Spergel,
Global texture and the microwave background, Phys. Rev. Lett., 64 (1990), 2736-2739.
doi: 10.1103/PhysRevLett.64.2736. |
[24] |
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. |
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