Attractors of Navier--Stokes systems and of parabolic equations, and estimates for their dimensions

A. V. Babin ; M. I. Vishik

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 3-15

Résumé

One investigates the problem of the existence of an attractor $\mathfrak A$ of the semigroup $S_t$ generated by the solutions of the nonlinear nonstationary equations $$ \frac{\partial u}{\partial t}=A(u),\quad u\mid_{t=0}=u_0(t);\qquad S_tu_0\equiv u(t). $$ One proves a very general theorem on the existence of an attractor $\mathfrak A$ of the semigroup $S_t$ for $t\to\infty$. One gives examples of differential equations having attractors: a second-order quasilinear parabolic equation, a two-dimensional Navier–Stokes system, a monotone parabolic equation of any order. One proves a theorem on the finiteness of the Hausdorff dimension of the attractor $\mathfrak A$. One gives an estimate for the Hausdorff dimension of the attractor $\mathfrak A$ for a two-dimensional Navier–Stokes system.