Аннотация:
Цель этой статьи заключается в изложении результатов, касающихся математического
изучения статистической эволюции решений уравнений Навье–Стокса (см. [7] и особенно [8], [9]), которые могут служить для строгого математического обоснования теории турбулентности жидкости, ограниченной поверхностями. Изложение ведется на абстрактном функциональном языке (подсказанном работой [7]).
Koji Ohkitani, “Remarks on the principles of statistical fluid mechanics”, Phil. Trans. R. Soc. A., 380:2218 (2022)
Koji Ohkitani, “Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling”, Phys. Rev. E, 101:1 (2020)
A.C. Bronzi, C.F. Mondaini, R.M.S. Rosa, “Abstract framework for the theory of statistical solutions”, Journal of Differential Equations, 260:12 (2016), 8428
A.C.. Bronzi, C.F.. Mondaini, R.M.. S. Rosa, “Trajectory Statistical Solutions for Three-Dimensional Navier–Stokes-Like Systems”, SIAM J. Math. Anal, 46:3 (2014), 1893
Anne Bronzi, Ricardo Rosa, “On the convergence of statistical solutions of the 3D Navier–Stokes-α model as α vanishes”, DCDS-A, 34:1 (2013), 19
Themistoklis P. Sapsis, Pierre F.J. Lermusiaux, “Dynamically orthogonal field equations for continuous stochastic dynamical systems”, Physica D: Nonlinear Phenomena, 238:23-24 (2009), 2347
Themistoklis P. Sapsis, Gerassimos A. Athanassoulis, “New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation”, Probabilistic Engineering Mechanics, 23:2-3 (2008), 289
Dongho Chae, Namkwon Kim, “Homogeneous statistical solutions and the vanishing interfacial energy limit of the Cahn-Hilliard equation”, Nonlinear Analysis: Theory, Methods & Applications, 29:10 (1997), 1197
G. L. Sewell, NATO ASI Series, 324, On Three Levels, 1994, 11
Dongho Chae, “The vanishing viscosity limit of statistical solutions of the Navier–Stokes equations. I. 2-D periodic case”, Journal of Mathematical Analysis and Applications, 155:2 (1991), 437
Dongho Chae, “The vanishing viscosity limit of statistical solutions of the Navier–Stokes equations. II. The general case”, Journal of Mathematical Analysis and Applications, 155:2 (1991), 460
Taylan Alankus, “An exact representation of the space-time characteristic functional of turbulent Navier-Stokes flows with prescribed random initial states and driving forces”, J Stat Phys, 54:3-4 (1989), 859
А. А. Константинов, “К вопросу о функциональном подходе к проблеме турбулентности”, ТМФ, 42:1 (1980), 79–87; A. A. Konstantinov, “The functional approach to turbulence”, Theoret. and Math. Phys., 42:1 (1980), 52–58
S. Albeverio, M. Ribeiro de Faria, R. Høegh-Krohn, “Stationary measures for the periodic Euler flow in two dimensions”, J Statist Phys, 20:6 (1979), 585
М. И. Вишик, А. И. Комеч, А. В. Фурсиков, “Некоторые математические задачи статистической гидромеханики”, УМН, 34:5(209) (1979), 135–210; M. I. Vishik, A. I. Komech, A. V. Fursikov, “Some mathematical problems of statistical hydromechanics”, Russian Math. Surveys, 34:5 (1979), 149–234
G. Gallavotti, “Operatore di Liouville e soluzioni statistiche delle equazioni di Hamilton”, Annali di Matematica, 108:1 (1976), 227