The existence and asymptotic behavior of solutions to 3D viscous primitive equations with Caputo fractional time derivatives

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.