On Homogenization of Attractors to Two-Dimensional Navier – Stokes System in Periodic Layer with Small Obstacles
The work was supported by the Russian Science Foundation (project 25-11-00133)
DOI:
https://doi.org/10.52575/2687-0959-2026-58-2-147-156Keywords:
Navier – Stokes System, Trajectory Attractor, Homogenization, Domain Perforated in the Vicinity of the BoundaryAbstract
This paper considers a periodic problem for a two-dimensional Navier-Stokes system in a layer with small obstacles near the bottom. Assuming a no-slip condition at the obstacle boundary and a no-flow condition at the bottom, as well as a third boundary condition on the horizontal component, we investigate the asymptotic behavior of the trajectory attractors. Under these assumptions, we prove weak convergence of the trajectory attractors of this system to the trajectory attractors of the averaged Navier-Stokes system with a no-slip condition on the entire boundary.
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Список литературы
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Chechkin G.A., D’Apice C., De Maio U., Gadyl’shin RR. On Singularly Perturbed Steklov problem in Domain Perforated Along the Boundary. Comptes Rendus Me´canique. 2016;344(1):12–18.
Chechkin G.A., D’Apice C., De Maio U., Gadyl’shin RR. On the Steklov problem in a Domain Perforated Along a Part of the Boundary. Mode´lisation Mathe´matique et Analyse Nume´rique (M2AN). 2017;51(4)1317–1342. DOI: 10.1051/m2an/2016063
Chechkin G.A. The Meyers Estimates for Domains Perforated Along the Boundary. Mathematics. 2021;9(23):Art.No 3015.
Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., Goritsky A.Yu. Homogenization of Trajectory Attractors of 3D Navier–Stokes system with Randomly Oscillating Force. Discrete and Continuous Dynamical Systems. Series A (DCDS-A). 2017;37(5):2375–2393.
Bekmaganbetov K.A., Chechkin G.A., Toleubay A.M. Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium. Bull. Karaganda Univ. Math. Ser. 2022;3:35–50. DOI: 10.31489/2022M3/35-50
Chechkin G.A., Chepyzhov V.V., Pankratov L.S. Homogenization of Trajectory Attractors of Ginzburg–Landau equations with Randomly Oscillating Terms. Discrete and Continuous Dynamical Systems. Series B (DCDS-B). 2018;23(3):1133–1154.
Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V. Weak Convergence of Attractors of Reaction–Diffusion Systems with Randomly Oscillating Coefficients. Applicable Analysis. 2019;98(1-2):256–271.
Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V. “Strange Term” in Homogenization of Attractors of Reaction–Diffusion Equation in Perforated Domain. Chaos, Solitons & Fractals. 2020;140:Art.No 110208.
Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., Tolemis AA. Homogenization of attractors to Ginzburg–Landau equations in media with locally periodic obstacles: sub- and supercritical cases. Bull. Karaganda Univ. Math. Ser. 2024;(2):40–56.
Chepyzhov V.V. , Vishik M.I. Trajectory attractors for the 3D Navier-Stokes system and some generalizations, Top. Meth. Nonlin. Anal. J. Julius Schauder Center. 1996;8:217–243.
Chepyzhov V.V. , Vishik M.I. Evolution equations and their trajectory attractors. J. Math.Pures Appl. 1997;76(10):913–964.
References
Temam R. Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematics Series, 68. New York (NY)–Berlin–Heidelberg–London–Paris–Tokyo–Hong Kong–Barcelona–Budapest: Springer-Verlag; 1988. 500 p.
Babin AV., Vishik MI. Attractors of evolution equations. Amsterdam: North–Holland Publishing Co.; 1992. 300 p.
Chepyzhov VV., Vishik MI. Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, 49. Providence, RI: American Mathematical Society; 2002. 363 p.
Lobo M., Oleinik OA., Pe´rez ME., Shaposhnikova TA. On homogenization of solutions of boundary value problems in domains perforated along manifolds. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 1997;25(3–4):611–629.
Gadyl’shin RR., Koroleva YuO., Chechkin GA. On the Convergence of Solutions and Eigenelements of a Boundary Value Problem in a Domain Perforated Along the Boundary. Differential Equations. 2010;46(5):667–680.
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Chechkin GA., Koroleva YO., Persson L-E., Wall P. On Spectrum of the Laplacian in a Circle Perforated Along the Boundary: Application to a Friedrichs–Type Inequality. International Journal of Differential Equations. 2011;2011:Art.No 619623.
Chechkin GA., Koroleva YO., Persson L-E., Wall P. A new Weighted Friedrichs–type Inequality for a Perforated Domain with a Sharp Constant. Eurasian Mathematical Journal. 2011;2(1):81–103.
Amirat Y., Bodart O., Chechkin GA., Piatnitski AL. Asymptotics of a spectral-sieve problem. J. Math. Anal. Appl. 2016;435(2):1652–1671. DOI: 10.1016/j.jmaa.2015.11.014
Chechkin GA., D’Apice C., De Maio U., Gadyl’shin RR. On Singularly Perturbed Steklov problem in Domain Perforated Along the Boundary. Comptes Rendus Me´canique. 2016;344(1):12–18.
Chechkin GA., D’Apice C., De Maio U., Gadyl’shin RR. On the Steklov problem in a Domain Perforated Along a Part of the Boundary. Mode´lisation Mathe´matique et Analyse Nume´rique (M2AN). 2017;51(4)1317–1342. DOI: 10.1051/m2an/2016063
Chechkin G.A. The Meyers Estimates for Domains Perforated Along the Boundary. Mathematics. 2021;9(23):Art.No 3015.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV., Goritsky AYu. Homogenization of Trajectory Attractors of 3D Navier–Stokes system with Randomly Oscillating Force. Discrete and Continuous Dynamical Systems. Series A (DCDS-A). 2017;37(5):2375–2393.
Bekmaganbetov KA., Chechkin GA., Toleubay AM. Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium. Bull. Karaganda Univ. Math. Ser. 2022;3:35–50. DOI: 10.31489/2022M3/35-50
Chechkin GA., Chepyzhov VV., Pankratov LS. Homogenization of Trajectory Attractors of Ginzburg–Landau equations with Randomly Oscillating Terms. Discrete and Continuous Dynamical Systems. Series B (DCDS-B). 2018;23(3):1133–1154.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV. Weak Convergence of Attractors of Reaction–Diffusion Systems with Randomly Oscillating Coefficients. Applicable Analysis. 2019;98(1-2):256–271.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV. “Strange Term” in Homogenization of Attractors of Reaction–Diffusion Equation in Perforated Domain. Chaos, Solitons & Fractals. 2020;140:Art.No 110208.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV., Tolemis AA. Homogenization of attractors to Ginzburg–Landau equations in media with locally periodic obstacles: sub- and supercritical cases. Bull. Karaganda Univ. Math. Ser. 2024;(2):40–56.
Chepyzhov VV. , Vishik MI. Trajectory attractors for the 3D Navier-Stokes system and some generalizations, Top. Meth. Nonlin. Anal. J. Julius Schauder Center. 1996;8:217–243.
Chepyzhov VV. , Vishik MI. Evolution equations and their trajectory attractors. J. Math.Pures Appl. 1997;76(10):913–964.
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