Об усреднении аттракторов двумерной системы Навье – Стокса в периодическом слое с мелкими препятствиями
Работа выполнена при поддержке РНФ (проект 25-11-00133).
DOI:
https://doi.org/10.52575/2687-0959-2026-58-2-147-156Ключевые слова:
система Навье – Стокса, траекторный аттрактор, усреднение, область, перфорированная в окрестности границыАннотация
В работе рассмотрена периодическая задача для двумерной системы Навье – Стокса в слое с мелкими препятствиями около дна. Предполагая, что на границе препятствий выставлено условие прилипания, а на дне условие непротекания, а также третье краевое условие на горизонтальную компоненту, мы исследуем асимптотическое поведение траекторных аттракторов. В этих предположениях доказана слабая сходимость траекторных аттракторов этой системы к траекторным аттракторам усредненной системы Навье – Стокса с условием прилипания на всей границе.
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Список литературы
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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
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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.
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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.
Gadyl’shin RR., Koroleva YuO., Chechkin GA. On the Asymptotics of a Simple Eigenvalue to Boundary Value Problem in a Domain Perforated Along the Boundary. Differential Equations. 2011;47(6):822–831.
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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