May  2020, 19(5): 2419-2443. doi: 10.3934/cpaa.2020106

Strong convergence of trajectory attractors for reaction–diffusion systems with random rapidly oscillating terms

1. 

M.V. Lomonosov Moscow State University, Kazakhstan Branch, Kazhymukan st. 11, Astana, 010010, Kazakhstan

2. 

Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty, 050010, Kazakhstan

3. 

M.V. Lomonosov Moscow State University, Moscow, 119991, Russia

4. 

Institute of Mathematics with Computing Center. Subdivision of the Ufa, Federal Research Center of Russian Academy of Science, Chernyshevskogo st. 112, Ufa, 450008, Russia

5. 

Institute for Information Transmission Problems - Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russia

6. 

National Research University Higher School of Economics, Myasnitskaya Street 20, Moscow 101000, Russia

*Corresponding author

Received  May 2019 Revised  November 2019 Published  March 2020

Fund Project: The work is partially supported by the Russian Foundation of Basic Researches (GAC project 18-01-00046 and VVC project 17-01-00515) and Russian Science Foundation (project 20-11-20272). Work of KAB is supported in part by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan AP05132071

We consider reaction–diffusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diffusion system whose terms are the average of the corresponding terms of the original system. Special attention is given to the case when the convergence of random trajectory attractors holds in the strong topology.

Citation: Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov. Strong convergence of trajectory attractors for reaction–diffusion systems with random rapidly oscillating terms. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2419-2443. doi: 10.3934/cpaa.2020106
References:
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Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stoch. Process. Their Appl., 121 (2011), 1-23.  doi: 10.1016/j.spa.2010.08.011.  Google Scholar

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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North–Holland Publishing Co., Amsterdam, 1992; Nauka, Moscow, 1989.  Google Scholar

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N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and Its Applications (Soviet Series), Vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

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K. A. BekmaganbetovG. A. Chechkin and V. V. Chepyzhov, Homogenization of Random Attractors for Reaction–Diffusion Systems, C. R. Mec., 344 (2016), 753-758.  doi: 10.1016/j.crme.2016.10.015.  Google Scholar

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show all references

References:
[1]

Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stoch. Process. Their Appl., 121 (2011), 1-23.  doi: 10.1016/j.spa.2010.08.011.  Google Scholar

[2]

V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York, Amsterdam, 1968.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North–Holland Publishing Co., Amsterdam, 1992; Nauka, Moscow, 1989.  Google Scholar

[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and Its Applications (Soviet Series), Vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

[5]

J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[6]

K. A. BekmaganbetovG. A. ChechkinV. V. Chepyzhov and A. Yu. Goritsky, Homogenization of trajectory attractors of 3D Navier–Stokes system with Randomly Oscillating Force, Discrete Contin. Dyn. Syst., 37 (2017), 2375-2393.  doi: 10.3934/dcds.2017103.  Google Scholar

[7]

K. A. BekmaganbetovG. A. Chechkin and V. V. Chepyzhov, Homogenization of Random Attractors for Reaction–Diffusion Systems, C. R. Mec., 344 (2016), 753-758.  doi: 10.1016/j.crme.2016.10.015.  Google Scholar

[8]

A. Bensoussan, J. L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North–Holland, Amsterdam, 1978.  Google Scholar

[9]

G. D. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci. U. S. A., 17 (1931), 656-660.  doi: 10.1073/pnas.17.12.656.  Google Scholar

[10]

F. Boyer F. and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Vol. 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[11]

N. N. Bogolyubov and Ya. A. Mitropolski, Asymptotic Methods in the Theory of Non-linear Oscillations, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon & Breach Science Publishers, New York, 1961.  Google Scholar

[12]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.  Google Scholar

[13]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[14]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[15]

G. A. ChechkinT. P. ChechkinaC. D'Apice and U. De Maio, Homogenization in domains randomly perforated along the boundary, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 713-730.  doi: 10.3934/dcdsb.2009.12.713.  Google Scholar

[16]

G. A. ChechkinT. P. ChechkinaC. D'ApiceU. De Maio and T. A. Mel'nyk, Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission Zone, Appl. Anal., 88 (2009), 1543-1562.  doi: 10.1080/00036810902994268.  Google Scholar

[17]

G. A. ChechkinT. P. ChechkinaC. D'ApiceU. De Maio and T. A. Mel'nyk, Homogenization of 3D thick cascade junction with the random transmission zone periodic in one direction, Russ. J. Math. Phys., 17 (2010), 35-55.  doi: 10.1134/S1061920810010048.  Google Scholar

[18]

G. A. ChechkinC. D'ApiceU. De Maio and A. L. Piatnitski, On the rate of convergence of solutions in domain with random multilevel oscillating boundary, Asymptotic Anal., 87 (2014), 1-28.  doi: 10.3233/ASY-131194.  Google Scholar

[19]

G. A. ChechkinT. P. ChechkinaT. S. Ratiu and M. S. Romanov, Nematodynamics and random homogenization, Appl. Anal., 95 (2016), 2243-2253.  doi: 10.1080/00036811.2015.1036241.  Google Scholar

[20]

G. A. ChechkinV. V. Chepyzhov and L. S. Pankratov, Homogenization of trajectory attractors of Ginzburg–Landau equations with randomly oscillating terms, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1133-1154.  doi: 10.3934/dcdsb.2018145.  Google Scholar

[21]

V. V. ChepyzhovA. Yu. Goritski and M. I. Vishik, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005), 17-39.   Google Scholar

[22]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008), 469-491.  doi: 10.1016/j.matpur.2008.07.001.  Google Scholar

[23]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370.  doi: 10.1088/0951-7715/22/2/006.  Google Scholar

[24]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.  Google Scholar

[25]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[26]

M. I. Vishik and V. V. Chepyzhov, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[27]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, ESAIM Control Optim. Calc. Var., 8 (2002), 467-487.  doi: 10.1051/cocv:2002056.  Google Scholar

[28]

V. V. Chepyzhov and M. I. Vishik, Global attractors for non-autonomous Ginzburg-Landau equation with singularly oscillating terms, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 123-148.   Google Scholar

[29]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dyn. Differ. Equ., 19 (2007), 655-684.  doi: 10.1007/s10884-007-9077-y.  Google Scholar

[30]

V. V. ChepyzhovM. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27.  Google Scholar

[31]

I. D. Chueshov and B. Schmalfuß, Averaging of attractors and inertial manifolds for parabolic PDE With random coefficients, Adv. Nonlinear Stud., 5 (2005), 461-492.  doi: 10.1515/ans-2005-0402.  Google Scholar

[32]

M. Efendiev and S. Zelik, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 19 (2002), 961-989.  doi: 10.1016/S0294-1449(02)00115-4.  Google Scholar

[33]

M. Efendiev and S. Zelik, The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differ. Equ., 8 (2003), 673-732.   Google Scholar

[34]

B. Fiedler and M. I. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial sequences, Adv. Differ. Equ., 6 (2001), 1377-1408.   Google Scholar

[35]

B. Fiedler and M. I. Vishik, Quantitative homogenization of global attractors for reaction–diffusion systems with rapidly oscillating terms, Asymptotic Anal., 34 (2003), 159-185.   Google Scholar

[36]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differ. Equ., 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[37]

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, J. Integral Equ. Appl., 2 (1990), 463-494.  doi: 10.1216/jiea/1181075583.  Google Scholar

[38]

V. V. Jikov, S. M. Kozlov adnd O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[39]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillatingr ight-hand sides, Sb. Math., 187 (1996), 635-677.  doi: 10.1070/SM1996v187n05ABEH000126.  Google Scholar

[40]

A. A. Ilyin, Global averaging of dissipative dynamical systems, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 22 (1998), 165-191.   Google Scholar

[41] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow, 1956 [in Russia]. Pergamon Press, London, 1964.   Google Scholar
[42]

J. L. Lions and E. Magenes, Problemes oux Limites non Homogénes et Applications, Dunod, Gauthier-Villars, Paris, 1968.  Google Scholar

[43]

J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[44]

V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grain Boundary, Naukova Dumka, Kiev, 1974 [in Russian].  Google Scholar

[45]

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