# American Institute of Mathematical Sciences

November  2017, 16(6): 2357-2376. doi: 10.3934/cpaa.2017116

## Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions

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

* Corresponding author

Received  March 2017 Revised  April 2017 Published  July 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 first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.

Citation: Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
##### 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] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: American Mathematical Society (AMS), 2002. [4] 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. [5] 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. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Vol. 840, Springer-Verlag. Berlin-Heidelberg-New York, 1981. [7] A. Kostianko, Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions, submitted, 2015. [8] 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. [9] A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion advection systems. Part Ⅱ: periodic boundary conditions equations with periodic boundary conditions, submitted, 2017. [10] A. Kostianko and S. Zelik, Kwak transform and inertial manifolds revisited, submitted, 2017. [11] 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. [12] M. Kwak, Finite dimensional inertial forms for 2D Navier-Stokes equations, Indiana Univ. Math. J., 41 (1992), 927-982.  doi: 10.1512/iumj.1992.41.41051. [13] H. Kwean, An extension of the principle of spatial averaging for inertial manifolds, J. Aust. Math. Soc., Ser. A, 66 (1999), 125-142. [14] 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. [15] M. Miklavčič, A sharp condition for existence of an inertial manifold, J. Dyn. Differ. Equations, 3 (1991), 437-456.  doi: 10.1007/BF01049741. [16] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Amsterdam: Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [17] J. Robinson, Dimensions, Embeddings, and Attractors Cambridge: Cambridge University Press, 2011. [18] 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. [19] A. Romanov, Finite-dimensional limiting dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 99-112.  doi: 10.1070/SM2000v191n03ABEH000466. [20] A. Romanov, Finite-dimensional dynamics on attractors of nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001; translation from izv.  doi: 10.1070/IM2001v065n05ABEH000359. [21] A. Romanov, A parabolic equation with nonlocal diffusion without a smooth inertial manifold, Math. Notes, 96 (2014), 548-555.  doi: 10.1134/S0001434614090296. [22] A. Romanov, On the hyperbolicity properties of inertial manifolds of reaction-diffusion equation, Dynamics of PDEs, 13 (2016), 263-272.  doi: 10.4310/DPDE.2016.v13.n3.a4. [23] R. Rosa and R. Temam, Inertial manifolds and normal hyperbolicity, Acta Applicandae Mathematicae, 45 (1996), 1-50.  doi: 10.1007/BF00047882. [24] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, NY: Springer, 2nd ed. , 1997. doi: 10.1007/978-1-4612-0645-3. [25] J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on a circle, Nonlinearity, 21 (2008), 1533-1545.  doi: 10.1088/0951-7715/21/7/009. [26] J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Comm. Math. Phys., 285 (2009), 975-990.  doi: 10.1007/s00220-008-0460-2. [27] J. Vukadinovic, Global dissipativity and inertial manifolds for diffusive Burgers equations with low-wavenumber instability, Discrete Contin. Dyn. Syst., 29 (2011), 327-341.  doi: 10.3934/dcds.2011.29.327. [28] 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] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: American Mathematical Society (AMS), 2002. [4] 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. [5] 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. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Vol. 840, Springer-Verlag. Berlin-Heidelberg-New York, 1981. [7] A. Kostianko, Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions, submitted, 2015. [8] 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. [9] A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion advection systems. Part Ⅱ: periodic boundary conditions equations with periodic boundary conditions, submitted, 2017. [10] A. Kostianko and S. Zelik, Kwak transform and inertial manifolds revisited, submitted, 2017. [11] 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. [12] M. Kwak, Finite dimensional inertial forms for 2D Navier-Stokes equations, Indiana Univ. Math. J., 41 (1992), 927-982.  doi: 10.1512/iumj.1992.41.41051. [13] H. Kwean, An extension of the principle of spatial averaging for inertial manifolds, J. Aust. Math. Soc., Ser. A, 66 (1999), 125-142. [14] 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. [15] M. Miklavčič, A sharp condition for existence of an inertial manifold, J. Dyn. Differ. Equations, 3 (1991), 437-456.  doi: 10.1007/BF01049741. [16] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Amsterdam: Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [17] J. Robinson, Dimensions, Embeddings, and Attractors Cambridge: Cambridge University Press, 2011. [18] 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. [19] A. Romanov, Finite-dimensional limiting dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 99-112.  doi: 10.1070/SM2000v191n03ABEH000466. [20] A. Romanov, Finite-dimensional dynamics on attractors of nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001; translation from izv.  doi: 10.1070/IM2001v065n05ABEH000359. [21] A. Romanov, A parabolic equation with nonlocal diffusion without a smooth inertial manifold, Math. Notes, 96 (2014), 548-555.  doi: 10.1134/S0001434614090296. [22] A. Romanov, On the hyperbolicity properties of inertial manifolds of reaction-diffusion equation, Dynamics of PDEs, 13 (2016), 263-272.  doi: 10.4310/DPDE.2016.v13.n3.a4. [23] R. Rosa and R. Temam, Inertial manifolds and normal hyperbolicity, Acta Applicandae Mathematicae, 45 (1996), 1-50.  doi: 10.1007/BF00047882. [24] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, NY: Springer, 2nd ed. , 1997. doi: 10.1007/978-1-4612-0645-3. [25] J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on a circle, Nonlinearity, 21 (2008), 1533-1545.  doi: 10.1088/0951-7715/21/7/009. [26] J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Comm. Math. Phys., 285 (2009), 975-990.  doi: 10.1007/s00220-008-0460-2. [27] J. Vukadinovic, Global dissipativity and inertial manifolds for diffusive Burgers equations with low-wavenumber instability, Discrete Contin. Dyn. Syst., 29 (2011), 327-341.  doi: 10.3934/dcds.2011.29.327. [28] 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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