Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

M. I. Vishik ; V. V. Chepyzhov

Matematičeskie zametki, Tome 71 (2002) no. 2, pp. 194-213.

Résumé

We construct the trajectory attractor $\mathfrak A$ of a three-dimensional Navier–Stokes system with exciting force $g(x)\in H$. The set $\mathfrak A$ consists of a class of solutions to this system which are bounded in $H$, defined on the positive semi-infinite interval $\mathbb R_+$ of the time axis, and can be extended to the entire time axis $\mathbb R$ so that they still remain bounded-in-$H$ solutions of the Navier–Stokes system. In this case any family of bounded-in-$L_\infty (\mathbb R_+;H)$ solutions of this system comes arbitrary close to the trajectory attractor $\mathfrak A$. We prove that the solutions $\{u(x,t), t\ge 0\}$ are continuous in $t$ if they are treated in the space of functions ranging in $H^{-\delta }$, $0\delta \le 1$. The restriction of the trajectory attractor $\mathfrak A$ to $t=0$, $\mathfrak A|_{t=0}=:\mathscr A$, is called the global attractor of the Navier–Stokes system. We prove that the global attractor $\mathscr A$ thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as $m\to \infty $ the trajectory attractors $\mathfrak A_m$ and the global attractors $\mathscr A_m$ of the $m$-order Galerkin approximations of the Navier–Stokes system converge to the trajectory and global attractors $\mathfrak A$ and $\mathscr A$, respectively. Similar problems are studied for the cases of an exciting force of the form $g=g(x,t)$ depending on time $t$ and of an external force $g$ rapidly oscillating with respect to the spatial variables or with respect to time $t$.

Bibliographie

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

[2] Temam R., Navier–Stokes Equations and Nonlinear Functional Analysis, Soc. Industr. and Appl. Math., Philadelphia, 1983; 2nd ed., 1995

[3] Chepyzhov V. V., Vishik M. I., “Evolution equations and their trajectory attractors”, J. Math. Pures Appl., 76 (1997), 913–964 | MR | Zbl

[4] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | Zbl

[5] Temam R., “Infinite-Dimensional Dynamical Systems in Mechanics and Physics”, Appl. Math. Ser., 68, Springer-Verlag, New York, 1988 | MR | Zbl

[6] Hale J. K., “Asymptotic Behaviour of Dissipative Systems”, Math. Surveys Monographs, 25, Amer. Math. Soc., Providence, RI, 1988 | MR | Zbl

[7] Vishik M. I., Chepyzhov V. V., “Usrednenie traektornykh attraktorov evolyutsionnykh uravnenii s bystro ostsilliruyuschimi chlenami”, Matem. sb., 192:1 (2001), 13–50 | MR | Zbl

[8] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Fizmatgiz, M., 1961

[9] Lions J.-L., Quelques méthodes de résolutions des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 | Zbl

[10] Chepyzhov V. V., Vishik M. I., Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002 | Zbl

[11] Chepyzhov V. V., Vishik M. I., “Trajectory attractors for 2D Navier–Stokes systems and some generalizations”, Topol. Meth. Nonlin. Anal. J. Juliusz Schauder Center., 8 (1996), 217–243 | MR | Zbl

[12] Chepyzhov V. V., Vishik M. I., “Attractors of non-autonomous dynamical systems and their dimension”, J. Math. Pures Appl., 73:3 (1994), 279–333 | MR | Zbl

[13] Haraux A., Systèmes dynamiques dissipatifs et applications, Masson, Paris–Milan–Barcelona–Rome, 1991 | Zbl

[14] Ilin A. A., “Usrednenie dissipativnykh dinamicheskikh sistem s bystro ostsilliruyuschmi pravymi chastyami”, Matem. sb., 187:5 (1996), 15–58 | MR | Zbl