Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces

Abstract

The existence and structure of uniform attractors in V is proved for nonautonomous 2D Navier-stokes equations on bounded domain with a new class of external forces, termed normal in L 2 loc (R; H) (see Definition 3.1), which are translation bounded but not translation compact in L 2 loc (R; H). To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.

Global well-posedness and global attractor of fourth order semilinear parabolic equation

“…then integrating (7) with respect to t from 0 to t gives From J.u 0 / < d and (6), we have J.u m .0// < d for sufficiently large m, then (8) gives…”

“…For the classical results, we refer the reader to Temam [4] and Hale [5]. Qingfeng Ma, Shouhong Wang and Chengkui Zhong [6] developed the global attractor theory where they replaced the uniformly compact conditions in the existence theorems of global attractors by the so-called Condition (C), which is more suitable in applications to partial differential equations(see [6][7][8] and [9] for details). Rodríguez-Bernal [10] studied the following reaction-diffusion equation with nonlinear boundary conditions 8 < :…”

Attractors for nonautonomous two-dimensional space periodic Navier–Stokes equations

“…We apply the theorem which is proved in Ref. 11 to nonautonomous 2D space periodic Navier-Stokes equations and obtained the existence of a uniform attractor in D͑A͒ if external force f͑t͒ is normal in L loc 2 ͑R , V͒. The attractor attracts all bounded subsets of D͑A͒ in the norm of D͑A͒.…”

“…The necessary and sufficient conditions for the existence of a uniform attractor of the nonautonomous infinite-dimensional dynamical system are proved in Ref. 11. This method is an extension of the corresponding one on the autonomous framework ͑see Ref.…”

Pullback attractors of 2D Navier–Stokes equations with weak damping, distributed delay, and continuous delay

“…For more results of the well-posedness and long-time behavior of the 2D autonomous incompressible Navier-Stokes equations, such as the existence of global solutions, the existence of global attractors, Hausdorff dimension and inertial manifold approximation, we can refer to Ladyzhenskaya [1], Robinson [2], Sell and You [3], and Temam [4,5], Babin and Vishik [6], Carvalho, Langa and Robinson [7] and Chueshov [8]. Moreover, Caraballo et al [9][10][11] derived the existence of global attractor for 2D autonomous incompressible Navier-Stokes equation with delays; Chepyzhov and Vishik [12,13] investigated the long-time behavior and convergence of corresponding uniform (global) attractors for the 2D Navier-Stokes equation with singularly oscillating forces as the external force tend to steady state by virtue of linearization method and estimate the corresponding difference equations; Foias and Temam [14,15] gave a survey about the geometric properties of solutions and the connection between solutions, dynamical systems, and turbulence for Navier-Stokes equations, such as the existence of !-limit sets; Rosa [16], Hou and Li [17] obtained the existence of global (uniform) attractors for the 2D autonomous (non-autonomous) incompressible Navier-Stokes equations in some unbounded domain, respectively; Lu, Wu and Zhong [18] and Lu [19] proved the existence of uniform attractors for 2D non-autonomous incompressible Navier-Stokes equations with normal or less regular normal external force by establishing a new dynamical systems framework; Miranville and Wang [20] derived the attractors for non-autonomous nonhomogeneous Navier-Stokes equations.…”