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On stability of functional differential equations with rapidly oscillating coefficients
Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions
University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom |
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [
References:
[1] |
A. Babin and M. Vishik,
Attractors of Evolution Equations Amsterdam etc. : North-Holland, 1992. |
[2] |
T. Bridges, J. Pennant and S. Zelik,
Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations, Arch. Ration. Mech. Anal., 214 (2014), 671-716.
doi: 10.1007/s00205-014-0772-7. |
[3] |
A. Eden, S. Zelik and V. Kalantarov,
Counterexamples to regularity of Mañé projections in the theory of attractors, Russ. Math. Surv., 68 (2013), 199-226.
|
[4] |
C. Foias, G. Sell and R. Temam,
Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[5] |
A. Kostianko and S. Zelik,
Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions, Commun. Pure Appl. Anal., 14 (2015), 2069-2094.
doi: 10.3934/cpaa.2015.14.2069. |
[6] |
A. Kostianko and S. Zelik,
Inertial manifolds for 1D reaction-diffusion advection systems. Part I: Dirichlet and Neumann boundary conditions, Commun. Pure Appl. Anal., 16 (2017), 2357-2376.
doi: 10.3934/cpaa.2017116. |
[7] |
P. Kuchment,
Floquet Theory for Partial Differential Equations Operator Theory: Advances and Applications, Vol. 60, Birkhauser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8573-7. |
[8] |
I. Kukavica,
Fourier parametrization of attractors for dissipative equations in one space dimension, J. Dynam. Diff. Eqns, 15 (2003), 473-484.
doi: 10.1023/B:JODY.0000009744.13730.01. |
[9] |
H. Kwean,
An extension of the principle of spatial averaging for inertial manifolds, J. Aust. Math. Soc., Ser. A, 66 (1999), 125-142.
|
[10] |
J. Mallet-Paret and G. Sell,
Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.
doi: 10.2307/1990993. |
[11] |
J. Mallet-Paret, G. Sell and Z. Shao,
Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055.
doi: 10.1512/iumj.1993.42.42048. |
[12] |
M. Miklavčič,
A sharp condition for existence of an inertial manifold, J. Dyn. Differ. Equations, 3 (1991), 437-456.
doi: 10.1007/BF01049741. |
[13] |
J. Robinson,
Dimensions, Embeddings, and Attractors Cambridge: Cambridge University Press, 2011. |
[14] |
A. Romanov,
Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47.
doi: 10.1070/IM1994v043n01ABEH001557. |
[15] |
A. Romanov,
Finite-dimensional limiting dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 99-112.
doi: 10.1070/SM2000v191n03ABEH000466. |
[16] |
A. Romanov,
Three counterexamples in the theory of inertial manifolds, Math. Notes, 68 (2000), 378-385.
doi: 10.1007/BF02674562. |
[17] |
A. Romanov,
Finite-dimensional dynamics on attractors of nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001; translation from izv.
doi: 10.1070/IM2001v065n05ABEH000359. |
[18] |
A. Romanov,
A parabolic equation with nonlocal diffusion without a smooth inertial manifold, Math. Notes, 96 (2014), 548-555.
doi: 10.1134/S0001434614090296. |
[19] |
R. Temam,
Infinite-dimensional Dynamical Systems in Mechanics and Physics 2nd ed. , New York, NY: Springer, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[20] |
S. Zelik,
Inertial manifolds and finite-dimensional reduction for dissipative pdes, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1245-1327.
doi: 10.1017/S0308210513000073. |
show all references
References:
[1] |
A. Babin and M. Vishik,
Attractors of Evolution Equations Amsterdam etc. : North-Holland, 1992. |
[2] |
T. Bridges, J. Pennant and S. Zelik,
Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations, Arch. Ration. Mech. Anal., 214 (2014), 671-716.
doi: 10.1007/s00205-014-0772-7. |
[3] |
A. Eden, S. Zelik and V. Kalantarov,
Counterexamples to regularity of Mañé projections in the theory of attractors, Russ. Math. Surv., 68 (2013), 199-226.
|
[4] |
C. Foias, G. Sell and R. Temam,
Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[5] |
A. Kostianko and S. Zelik,
Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions, Commun. Pure Appl. Anal., 14 (2015), 2069-2094.
doi: 10.3934/cpaa.2015.14.2069. |
[6] |
A. Kostianko and S. Zelik,
Inertial manifolds for 1D reaction-diffusion advection systems. Part I: Dirichlet and Neumann boundary conditions, Commun. Pure Appl. Anal., 16 (2017), 2357-2376.
doi: 10.3934/cpaa.2017116. |
[7] |
P. Kuchment,
Floquet Theory for Partial Differential Equations Operator Theory: Advances and Applications, Vol. 60, Birkhauser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8573-7. |
[8] |
I. Kukavica,
Fourier parametrization of attractors for dissipative equations in one space dimension, J. Dynam. Diff. Eqns, 15 (2003), 473-484.
doi: 10.1023/B:JODY.0000009744.13730.01. |
[9] |
H. Kwean,
An extension of the principle of spatial averaging for inertial manifolds, J. Aust. Math. Soc., Ser. A, 66 (1999), 125-142.
|
[10] |
J. Mallet-Paret and G. Sell,
Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.
doi: 10.2307/1990993. |
[11] |
J. Mallet-Paret, G. Sell and Z. Shao,
Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055.
doi: 10.1512/iumj.1993.42.42048. |
[12] |
M. Miklavčič,
A sharp condition for existence of an inertial manifold, J. Dyn. Differ. Equations, 3 (1991), 437-456.
doi: 10.1007/BF01049741. |
[13] |
J. Robinson,
Dimensions, Embeddings, and Attractors Cambridge: Cambridge University Press, 2011. |
[14] |
A. Romanov,
Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47.
doi: 10.1070/IM1994v043n01ABEH001557. |
[15] |
A. Romanov,
Finite-dimensional limiting dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 99-112.
doi: 10.1070/SM2000v191n03ABEH000466. |
[16] |
A. Romanov,
Three counterexamples in the theory of inertial manifolds, Math. Notes, 68 (2000), 378-385.
doi: 10.1007/BF02674562. |
[17] |
A. Romanov,
Finite-dimensional dynamics on attractors of nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001; translation from izv.
doi: 10.1070/IM2001v065n05ABEH000359. |
[18] |
A. Romanov,
A parabolic equation with nonlocal diffusion without a smooth inertial manifold, Math. Notes, 96 (2014), 548-555.
doi: 10.1134/S0001434614090296. |
[19] |
R. Temam,
Infinite-dimensional Dynamical Systems in Mechanics and Physics 2nd ed. , New York, NY: Springer, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[20] |
S. Zelik,
Inertial manifolds and finite-dimensional reduction for dissipative pdes, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1245-1327.
doi: 10.1017/S0308210513000073. |
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