# American Institute of Mathematical Sciences

January  2018, 17(1): 285-317. doi: 10.3934/cpaa.2018017

## Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

 University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom

* Corresponding author

Received  March 2017 Published  September 2017

Fund Project: This work is partially supported by the grants 14-41-00044 and 14-21-00025 of RSF as well as grants 14-01-00346 and 15-01-03587 of RFBR.

This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [6] and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed with periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.

Citation: Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure and Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
##### 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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##### 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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