Attractors for Nonautonomous Navier–Stokes System and Other Partial Differential Equations

Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing

“…In this section we study a 2D Navier-Stokes equation as an example to apply our theoretical analysis. First, let us introduce translation compact/bounded functions which are known important in the study of uniform attractors [10,8,9,29]. From now on, we fix g(t, x) ∈ L 2 loc (R; (L 2 (O)) 2 ) to be translation bounded.…”

“…In this part, we show a determining modes result and finite dimensionality of the D-random cocycle attractor for the Navier-Stokes equation. The finite dimensionality of the attractor (kernel sections) for non-autonomous Navier-Stokes equations and other models goes back to the works of Chepyzhov and Vishik (see, for instance, [8,9,5,48,7]). Note that our method does not give the better upper bounds on the dimension of attractors for the non-autonomous Navier-Stokes, as the Lyapunov method (see, for instance, [6,10,40]).…”

“…What is worth a note is that the finite dimensionality of cocycle attractors obtained in this paper does hold uniformly for all components of the attractor, that is, the upper bound of the dimension is uniform for all components. Applying to a 2D Navier-Stokes equation we find that the uniformity of the bound is supported by the translation-boundedness of the external force of the system ( [10,8,9]), which is known often a sufficient condition for the existence of a random uniform attractor [29].…”

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

“…⋄ Nonlinear wave equations, 26,27,29,30,35 ⋄ Nonlinear Schrödinger equation, 28 ⋄ Navier-Stokes equation, 36 ⋄ Ginzburg-Landau equation. 37…”

“…To prove Theorem 1.3, we need to borrow some ideas developed in Chepyzhov et al [35][36][37] ■ Case 1.…”