May  2017, 37(5): 2375-2393. doi: 10.3934/dcds.2017103

Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force

1. 

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

2. 

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

3. 

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

4. 

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

* Corresponding author: G.A.Chechkin

Received  January 2016 Revised  December 2016 Published  February 2017

Fund Project: Work of KAB is partially supported by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (grant no. 0816/GF4). Work of GAC, VVC, and AYG is partially supported by the Russian Foundation for Basic Research (projects 15-01-07920 and 14-01-00346)

We consider the 3D Navier-Stokes systems with randomly rapidly oscillating right-hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier-Stokes systems whose right-hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier-Stokes systems is uniquely solvable.

Bibliography: 44 titles.

Citation: Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103
References:
[1]

Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stochastic Processes and their Applications, 121 (2011), 1-23. 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. Google Scholar
[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989. Google Scholar

[5]

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

[6]

A. Bensoussan, J. -L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures Corrected reprint of the 1978 original [MR0503330]. AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374. Google Scholar

[7]

G. D. Birkhoff, Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660. doi: 10.1073/pnas.17.2.656. Google Scholar

[8]

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. doi: 10.1063/1.3050754. Google Scholar

[9]

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

[10]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 234. American Mathematical Society, Providence, RI, 2007. doi: 978-0-8218-3873-0; 0-8218-3873-3. Google Scholar

[11]

G. A. ChechkinT. P. ChechkinaC. D'Apice and U. De Maio, Homogenization in Domains Randomly Perforated Along the Boundary, Discrete Continuous Dynam. Systems -B, 12 (2009), 713-730. doi: 10.3934/dcdsb.2009.12.713. Google Scholar

[12]

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, Applicable Analysis, 88 (2009), 1543-1562. doi: 10.1080/00036810902994268. Google Scholar

[13]

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. Google Scholar

[14]

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 Analysis, 87 (2014), 1-28. Google Scholar

[15]

G. A. ChechkinT. P. ChechkinaT. S. Ratiu and M. S. Romanov, Nematodynamics and Random Homogenization, Applicable Analysis, 95 (2016), 2243-2253. doi: 10.1080/00036811.2015.1036241. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

[0-8218-2950-5]V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32] V. V. JikovS. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar
[33]

J. -L. Lions, Quelques Méthodes de Résolutions Des Problémes Aux Limites non Linéaires Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[34]

V. A. Marchenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei (Russian) [Boundary value problems in domains with a fine-grained boundary] Naukova Dumka, Kiev, 1974. Google Scholar

[35]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc. , Boston, MA, 2006. doi: 978-0-8176-4351-5; 0-8176-4351-6. Google Scholar

[36]

L. S. Pankratov and I. D. Cheushov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352. Google Scholar

[37]

E. Sánchez-Palencia, Homogenization Techniques for Composite Media Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. doi: 3-540-17616-0. Google Scholar

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 0-387-94866-X. Google Scholar

[39]

M. I. Vishik and V. V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47. doi: 10.1070/SM2001v192n01ABEH000534. Google Scholar

[40]

M. I. Vishik and V. V. Chepyzhov, Approximation of trajectories lying on a global attractor of a hyperbolic equation with an exterior force that oscillates rapidly over time, Sb. Math., 194 (2003), 1273-1300. Google Scholar

[41]

M. I. Vishik and V. V. Chepyzhov, Attractors of dissipative hyperbolic equations with singularly oscillating external forces, Math. Notes, 79 (2006), 483-504. doi: 10.1007/s11006-006-0054-2. Google Scholar

[42]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753. Google Scholar

[43]

M. I. Vishik and B. Fiedler, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russian Math. Surveys, 57 (2002), 709-728. Google Scholar

[44]

S. Zelik, Global averaging and parametric resonances in damped semilinear wave equations, Proc. Roy. Soc. Edinburgh Sect.A, 136 (2006), 1053-1097. doi: 10.1017/S0308210500004881. Google Scholar

show all references

References:
[1]

Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stochastic Processes and their Applications, 121 (2011), 1-23. 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. Google Scholar
[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989. Google Scholar

[5]

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

[6]

A. Bensoussan, J. -L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures Corrected reprint of the 1978 original [MR0503330]. AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374. Google Scholar

[7]

G. D. Birkhoff, Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660. doi: 10.1073/pnas.17.2.656. Google Scholar

[8]

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. doi: 10.1063/1.3050754. Google Scholar

[9]

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

[10]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 234. American Mathematical Society, Providence, RI, 2007. doi: 978-0-8218-3873-0; 0-8218-3873-3. Google Scholar

[11]

G. A. ChechkinT. P. ChechkinaC. D'Apice and U. De Maio, Homogenization in Domains Randomly Perforated Along the Boundary, Discrete Continuous Dynam. Systems -B, 12 (2009), 713-730. doi: 10.3934/dcdsb.2009.12.713. Google Scholar

[12]

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, Applicable Analysis, 88 (2009), 1543-1562. doi: 10.1080/00036810902994268. Google Scholar

[13]

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. Google Scholar

[14]

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 Analysis, 87 (2014), 1-28. Google Scholar

[15]

G. A. ChechkinT. P. ChechkinaT. S. Ratiu and M. S. Romanov, Nematodynamics and Random Homogenization, Applicable Analysis, 95 (2016), 2243-2253. doi: 10.1080/00036811.2015.1036241. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

[0-8218-2950-5]V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32] V. V. JikovS. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar
[33]

J. -L. Lions, Quelques Méthodes de Résolutions Des Problémes Aux Limites non Linéaires Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[34]

V. A. Marchenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei (Russian) [Boundary value problems in domains with a fine-grained boundary] Naukova Dumka, Kiev, 1974. Google Scholar

[35]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc. , Boston, MA, 2006. doi: 978-0-8176-4351-5; 0-8176-4351-6. Google Scholar

[36]

L. S. Pankratov and I. D. Cheushov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352. Google Scholar

[37]

E. Sánchez-Palencia, Homogenization Techniques for Composite Media Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. doi: 3-540-17616-0. Google Scholar

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 0-387-94866-X. Google Scholar

[39]

M. I. Vishik and V. V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47. doi: 10.1070/SM2001v192n01ABEH000534. Google Scholar

[40]

M. I. Vishik and V. V. Chepyzhov, Approximation of trajectories lying on a global attractor of a hyperbolic equation with an exterior force that oscillates rapidly over time, Sb. Math., 194 (2003), 1273-1300. Google Scholar

[41]

M. I. Vishik and V. V. Chepyzhov, Attractors of dissipative hyperbolic equations with singularly oscillating external forces, Math. Notes, 79 (2006), 483-504. doi: 10.1007/s11006-006-0054-2. Google Scholar

[42]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753. Google Scholar

[43]

M. I. Vishik and B. Fiedler, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russian Math. Surveys, 57 (2002), 709-728. Google Scholar

[44]

S. Zelik, Global averaging and parametric resonances in damped semilinear wave equations, Proc. Roy. Soc. Edinburgh Sect.A, 136 (2006), 1053-1097. doi: 10.1017/S0308210500004881. Google Scholar

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Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

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