Abstract
We consider the existence of <svg style="vertical-align:-2.3205pt;width:43.825001px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 43.825001 15.0875" width="43.825001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D449" d="M730 650l-8 -28q-52 -4 -72 -18t-64 -77q-79 -113 -321 -539h-33l-119 541q-13 59 -29.5 73t-66.5 20l7 28h245l-8 -28l-28 -5q-33 -6 -40.5 -15.5t-0.5 -38.5l102 -450h2q191 320 246 430q21 42 15.5 56t-43.5 19l-31 4l7 28h240z" /></g><g transform="matrix(.017,-0,0,-.017,18.557,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,25.255,12.138)"><use xlink:href="#x1D449"/></g><g transform="matrix(.017,-0,0,-.017,37.868,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>-pullback attractor for nonautonomous primitive equations of large-scale ocean and atmosphere dynamics in a three-dimensional bounded cylindrical domain by verifying pullback <svg style="vertical-align:-0.20474pt;width:15.1625px;" id="M2" height="12.0625" version="1.1" viewBox="0 0 15.1625 12.0625" width="15.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x1D49F" d="M810 587l8 -20q-26 -12 -38 -20q-10 -7 -33 -33q57 -88 57 -190q0 -140 -99 -240q-97 -99 -223 -99q-70 0 -156 33q-75 -27 -161 -27q-61 0 -95 17t-34 46q0 25 30 40t77 15q72 0 170 -52q81 49 203 246q82 134 179 216q-42 59 -121 98q-78 38 -159 38q-145 0 -225 -64
q-80 -65 -80 -169q0 -60 46 -103q46 -42 118 -42q99 0 157 66.5t59 194.5h19q2 -20 2 -43q0 -62 -22.5 -112.5t-60.5 -78.5q-74 -57 -152 -57q-89 0 -148 50q-59 51 -59 133q0 110 97 183q98 74 242 74q95 0 178.5 -38.5t138.5 -105.5q37 27 85 44zM756 336q0 77 -37 144
q-44 -61 -103 -178q-75 -152 -196 -234q-24 -16 -53 -32q60 -20 111 -20q114 0 196 87q82 88 82 233zM276 39q-74 40 -132 40q-67 0 -67 -25q0 -33 87 -33q66 0 112 18z" /></g> </svg> condition.
Highlights
This paper is concerned with the existence of pullback attractor for the following nonautonomous primitive equations of large-scale ocean and atmosphere dynamics: ∂V ∂t + (V ⋅ ∇) V − z (∫ −h ∇V (x, y, ζ, t) dζ)
It is clear that u satisfies the following equation obtained by differentiating (1) with respect to z:
We prove that the process {U(t, τ)}t≥τ associated with (1)–(5) is pullback asymptotically compact in
Summary
This paper is concerned with the existence of pullback attractor for the following nonautonomous primitive equations of large-scale ocean and atmosphere dynamics:. The existence and uniqueness of global strong solutions for the initial boundary value problem of the threedimensional viscous primitive equations of large-scale ocean were established by the authors in [14]. In [20], the authors considered the long-time dynamics of the primitive equations of large-scale atmosphere and obtained a weakly compact global attractor A which captures all the trajectories with respect to V-weak topology. In [30], the authors proved the existence of (V, V)-global attractor for the primitive equations of large-scale atmosphere and ocean dynamics by use of the Aubin-Lions compactness theorem. Throughout this paper, let X be a Banach space endowed with the norm ‖ ⋅ ‖X, let ‖u‖p be the Lp(Ω)-norm of u, and let C be positive constants, which may be different from line to line
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