Pullback Attractor for Nonautonomous Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics

Abstract

We consider the existence of (,)-pullback attractor for nonautonomous primitive equations of large-scale ocean and atmosphere dynamics in a three-dimensional bounded cylindrical domain by verifying pullback D condition.

Pullback attractor for the three dimensional nonautonomous primitive equations of large‐scale ocean and atmosphere dynamics

“…Here, we prove the pullback -asymptotical compactness in (H 2 (Ω)) 3 ∩ V of the process {U(t, )} t ≥ corresponding to problem (7) to (9) to obtain the existence of a pullback -attractor in (H 2 (Ω)) 3 ∩ V. From (19), (26), and (40) to (42), we immediately obtain the following results.…”

“…The long-time behavior of solutions of such equations have been studied extensively in recent years. [22][23][24][25][26][27][28][29][30] In particular, the existence of a pullback attractor in L p (Ω) for the reaction-diffusion equation was obtained by Li et al 27 under the following assumption:…”

“…for all t ∈ R, where is some positive constant. The existence of a pullback attractor in V for the process {U(t, )} t ≥ associated with the three-dimensional nonautonomous primitive equations of large-scale ocean and atmosphere dynamics was established by Li and Li 26 by verifying the pullback -condition. Mejo 18,19 has proved the existence of a uniform attractor for the three-dimensional nonautonomous primitive equations with oscillating external force and investigated its structure and the upper semicontinuous properties.…”

“…Nonautonomous equations appear in many applications of the natural sciences, so they are of great importance and interest. The long‐time behavior of solutions of such equations have been studied extensively in recent years . In particular, the existence of a pullback attractor in L p (Ω) for the reaction‐diffusion equation was obtained by Li et al under the following assumption: for any some M >0, and any α ∈(0, λ 1 ), where λ 1 is the first eigenvalue of −Δ with Dirichlet boundary condition.…”

“…Yang et al have proved the existence of a pullback attractor for the nonautonomous quasi‐linear parabolic equations with a dynamical boundary condition by using a new type of uniform Grönwall inequality under the assumptions that the nonlinearities satisfy the polynomial growth conditions with some fixed orders and the external force satisfies some weak assumption: for all , where σ is some positive constant. The existence of a pullback attractor in V for the process { U ( t , τ )} t ≥ τ associated with the three‐dimensional nonautonomous primitive equations of large‐scale ocean and atmosphere dynamics was established by Li and Li by verifying the pullback ‐condition. Mejo has proved the existence of a uniform attractor for the three‐dimensional nonautonomous primitive equations with oscillating external force and investigated its structure and the upper semicontinuous properties.…”