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Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces
Reducibility of quasi-periodically forced circle flows
1. | Department of Mathematics, University of Mississippi, Hume Hall 305, University, MS 38677, USA |
2. | Departamento de Matemática, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal |
We develop a renormalization group approach to the problem of reducibility of quasi-periodically forced circle flows. We apply the method to prove a reducibility theorem for such flows.
References:
[1] |
V. I. Arnol'd,
Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25 (1961), 21-86.
|
[2] |
V. I. Arnol'd,
Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40.
|
[3] |
A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262. |
[4] |
A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239. |
[5] |
L. Corsi and G. Gentile,
Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316 (2012), 489-529.
doi: 10.1007/s00220-012-1548-2. |
[6] |
G. Gentile,
Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.
doi: 10.1007/s10955-006-9127-6. |
[7] |
G. Gentile,
Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.
doi: 10.1017/S0143385706000757. |
[8] |
M.-R. Herman,
Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49 (1979), 5-233.
|
[9] |
R. Johnson and J. Moser,
The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.
doi: 10.1007/BF01208484. |
[10] |
K. Khanin, J. Lopes Dias and J. Marklof,
Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.
doi: 10.1007/s00220-006-0125-y. |
[11] |
H. Koch and S. Kocić,
Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28 (2008), 1559-1585.
doi: 10.1017/S0143385707000892. |
[12] |
H. Koch and S. Kocić,
A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30 (2010), 1131-1146.
doi: 10.1017/S014338570900042X. |
[13] |
H. Koch and S. Kocić,
A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249 (2010), 1986-2004.
doi: 10.1016/j.jde.2010.05.004. |
[14] |
S. Kocić,
Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544.
doi: 10.1088/0951-7715/18/6/006. |
[15] |
S. Kocić,
Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29 (2011), 261-283.
doi: 10.3934/dcds.2011.29.261. |
[16] |
A. N. Kolmogorov,
On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.
|
[17] |
J. Lopes Dias,
A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.
doi: 10.1016/j.jde.2006.07.021. |
[18] |
J. Lopes Dias,
Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.
doi: 10.1016/j.jde.2008.04.006. |
[19] |
J. Moser,
Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.
doi: 10.1007/BF01399536. |
[20] |
H. Rüssmann,
On the one-dimensional Schrödinger equation with a quasiperiodic potential, Nonlinear Dynamics, Ann. New York Acad. Sci., New York Acad. Sci., New York, 357 (1980), 90-107.
|
[21] |
J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). |
[22] |
J.-C. Yoccoz,
Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors, Lecture Notes in Mathematics, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1784 (2002), 125-173.
doi: 10.1007/978-3-540-47928-4_3. |
show all references
References:
[1] |
V. I. Arnol'd,
Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25 (1961), 21-86.
|
[2] |
V. I. Arnol'd,
Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40.
|
[3] |
A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262. |
[4] |
A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239. |
[5] |
L. Corsi and G. Gentile,
Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316 (2012), 489-529.
doi: 10.1007/s00220-012-1548-2. |
[6] |
G. Gentile,
Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.
doi: 10.1007/s10955-006-9127-6. |
[7] |
G. Gentile,
Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.
doi: 10.1017/S0143385706000757. |
[8] |
M.-R. Herman,
Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49 (1979), 5-233.
|
[9] |
R. Johnson and J. Moser,
The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.
doi: 10.1007/BF01208484. |
[10] |
K. Khanin, J. Lopes Dias and J. Marklof,
Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.
doi: 10.1007/s00220-006-0125-y. |
[11] |
H. Koch and S. Kocić,
Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28 (2008), 1559-1585.
doi: 10.1017/S0143385707000892. |
[12] |
H. Koch and S. Kocić,
A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30 (2010), 1131-1146.
doi: 10.1017/S014338570900042X. |
[13] |
H. Koch and S. Kocić,
A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249 (2010), 1986-2004.
doi: 10.1016/j.jde.2010.05.004. |
[14] |
S. Kocić,
Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544.
doi: 10.1088/0951-7715/18/6/006. |
[15] |
S. Kocić,
Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29 (2011), 261-283.
doi: 10.3934/dcds.2011.29.261. |
[16] |
A. N. Kolmogorov,
On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.
|
[17] |
J. Lopes Dias,
A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.
doi: 10.1016/j.jde.2006.07.021. |
[18] |
J. Lopes Dias,
Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.
doi: 10.1016/j.jde.2008.04.006. |
[19] |
J. Moser,
Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.
doi: 10.1007/BF01399536. |
[20] |
H. Rüssmann,
On the one-dimensional Schrödinger equation with a quasiperiodic potential, Nonlinear Dynamics, Ann. New York Acad. Sci., New York Acad. Sci., New York, 357 (1980), 90-107.
|
[21] |
J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). |
[22] |
J.-C. Yoccoz,
Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors, Lecture Notes in Mathematics, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1784 (2002), 125-173.
doi: 10.1007/978-3-540-47928-4_3. |
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