Pullback Exponential Attractor For A Kind Of Non-autonomous Geophysical Equation

In order to understand the long-term weather forecast and the mechanism of climate change,people began to study the models of governing atmospheric and ocean flow: the primitive equations of the coupled atmosphere and ocean.In the past few decades,many mathematicians have been working on the well-posedness of primitive equations of the atmosphere and ocean coupling.The long-time behavior of dynamical systems is an important and challenging problem,since it can provide useful information on the future evolution of the system.The dynamic system is divided into autonomous system and nonautonomous systems.The long-time behaviour of dissipative dynamical systems generated by non-autonomous equations of physics can be described in terms of the so-called pullback attractors,which pullback attract arbitrary bounded subset in the phase space.The main objective of this paper is concerned with the existence of a pullback exponential attractor for the three dimensional non-autonomous primitive equations of large-scale ocean and atmosphere dynamics.Under some slightly strong assumptions on the heat source,the existence of a pullback attractor in a regular phase space V is proved by pullback asymptotical a priori estimates,and then prove the existence of pullback exponential attractor in the V,and provide the upper bound of the fractal dimension of the pullback exponential attractor in the phase space V by calculating the Lyapunov exponent.