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Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data
Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics
West University of Timişoara, Faculty of Mathematics and Computer Science, Department of Mathematics, Blvd. Vasile Pȃrvan, No. 4, 300223–Timişoara, Romȃnia |
Given an arbitrary fixed closed subset $ \mathcal{C}\subset\mathbb{R}^n $, we provide an explicit method to construct a dynamical system which admits the regular part of $ \mathcal{C} $ as globally bp-attracting set, i.e. a closed and invariant set which attracts every bounded positive orbit of the dynamical system. As application, we provide an explicit method of leafwise asymptotic bp-stabilization of the regular part of an a-priori given invariant set of a conservative system. The theoretical results are illustrated for the completely integrable case of the Rössler dynamical system.
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J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.
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R. M. Tudoran and A. G\^irban, On the completely integrable case of the Rössler system, J. Math. Phys., 53 (2012), 052701, 10 pp.
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R. M. Tudoran,
Affine distributions on Riemannian manifolds with applications to dissipative dynamics, J. Geom. Phys., 92 (2015), 55-68.
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R. M. Tudoran,
Asymptotic bp-stabilization of a given closed invariant set of a smooth dynamical system, J. Differential Equations, 267 (2019), 3768-3777.
doi: 10.1016/j.jde.2019.04.013. |
show all references
References:
| [1] |
G.-I. Bischi, C. Mira and L. Gardini,
Unbounded sets of attraction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1437-1469.
doi: 10.1142/S0218127400000980. |
| [2] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
| [3] |
B. Günther and J. Segal,
Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.
doi: 10.1090/S0002-9939-1993-1170545-4. |
| [4] |
B. Günther,
Construction of differentiable flows with prescribed attractor, Topology Appl., 62 (1995), 87-91.
doi: 10.1016/0166-8641(94)00047-7. |
| [5] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, 1988. |
| [6] |
P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38. SIAM, Philadelphia, PA, 2002.
doi: 10.1137/1.9780898719222. |
| [7] |
J. M. Lee, Introduction to Smooth Manifolds, Second edition, Graduate Texts in Mathematics, 218. Springer, New York, 2013. |
| [8] |
T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra,
Chapter II: A crash course in geometric mechanics, London Math. Soc. Lecture Note Ser., Geometric mechanics and symmetry, Cambridge Univ. Press, Cambridge, 306 (2005), 23-156.
doi: 10.1017/CBO9780511526367.003. |
| [9] |
J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.
|
| [10] |
R. M. Tudoran and A. G\^irban, On the completely integrable case of the Rössler system, J. Math. Phys., 53 (2012), 052701, 10 pp.
doi: 10.1063/1.4708621. |
| [11] |
R. M. Tudoran,
Affine distributions on Riemannian manifolds with applications to dissipative dynamics, J. Geom. Phys., 92 (2015), 55-68.
doi: 10.1016/j.geomphys.2015.01.017. |
| [12] |
R. M. Tudoran,
Asymptotic bp-stabilization of a given closed invariant set of a smooth dynamical system, J. Differential Equations, 267 (2019), 3768-3777.
doi: 10.1016/j.jde.2019.04.013. |
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