Moment theory for the Navier--Stokes equations with a random right side

A. V. Fursikov

Izvestiya. Mathematics , Tome 41 (1993) no. 3, pp. 515-555.

Résumé

A theory of the moments of a statistical solution of the Navier–Stokes equations is constructed. The Cauchy problem for an infinite chain of equations which these moments satisfy is written out. Uniqueness of the solution of this Cauchy problem in appropriate function spaces is proved. The solution is not assumed to be positive definite, i.e., it may not be a collection of moments of the statistical solution. The concept of a statistical solution of the Navier-Stokes equations with a random right side is introduced, an equation for the statistical solution is derived, and the connection between this equation and the chain of moment equations is established. The problem of closure of the chain of moment equations is solved in the case of large Reynolds numbers, i.e., a sequence of extremal problems $\mathfrak A^N$ is constructed such that 1) the number of desired functions $M^N=\{M^N_{k,n}\}$ of the extremal problem $\mathfrak A^N$ is finite (and equal to $(N+1)N/2$), and 2) the solution $M^N$ of the problem $\mathfrak A^N$ approximates the solution of the Cauchy problem for the chain of moment equations: $M^N\to M$ as $N\to\infty$. The results are used to solve the problem of closure of the chain of Friedman-Keller moment equations corresponding to the three-dimensional Navier-Stokes system with zero right side, for large Reynolds numbers.

Bibliographie

[1] Monin A. S. Yaglom A. M., Statisticheskaya gidromekhanika, v. 2, Nauka, M., 1967, 720 pp.

[2] Frost U., Turbulentnost. Printsipy i primeneniya, Mir, M., 1980

[3] Vishik M. I., Fursikov A. V., Matematicheskie zadachi statisticheskoi gidromekhaniki, Nauka, M., 1980, 271 pp. | MR

[4] Fursikov A. V., “O probleme zamykaniya tsepochki momentnykh uravnenii v sluchae bolshikh chisel Reinoldsa”, Neklassicheskie uravneniya i uravneniya smeshannogo tipa, In-t matematiki SO AN SSSR, Novosibirsk, 1990, 231–250 | MR | Zbl

[5] Fursikov A. V., “O edinstvennosti resheniya tsepochki momentnykh uravnenii, sootvetstvuyuschikh trekhmernoi sisteme Nave–Stoksa”, Matem. sb., 134(176):4(12) (1987), 472–495 | MR

[6] Vishik M. I., Komech A. I., “Ob uravneniyakh Kolmogorova, sootvetstvuyuschikh dvumernoi stokhasticheskoi sisteme Nave–Stoksa”, Tr. Mosk. matem. ob-va, 46, 1983, 3–43 | MR | Zbl

[7] Fursikov A. V., “Analiticheskie funktsionaly i odnoznachnaya razreshimost kvazilineinykh dissipativnykh sistem pri pochti vsekh nachalnykh usloviyakh”, Tr. Mosk. matem. ob-va, 49, 1986, 3–55 | MR

[8] Fursikov A. V., “Svoistva reshenii nekotorykh ekstremalnykh zadach, svyazannykh s sistemoi Nave–Stoksa”, Matem. sb., 118(160):3(7) (1982), 323–349 | MR | Zbl

[9] Tikhonov A. I., Arsenin V. Ya., Metody reshenii nekorrektnykh zadach, Nauka, M., 1979, 285 pp. | MR

[10] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva I. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[11] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971, 372 pp. | Zbl

[12] Fursikov A. V., “Problema zamykaniya tsepochek momentnykh uravnenii, sootvetstvuyuschikh trekhmernoi sisteme Nave–Stoksa v sluchae bolshikh chisel Reinoldsa”, DAN SSSR, 319:1 (1991), 83–87 | MR | Zbl