Pullback exponential attractors for 3D non-autonomous Boussinesq equations with fractional Laplacian

Focusing on the pullback asymptotic behavior exhibited by the system, this paper studies the three-dimensional (3D) non-autonomous Boussinesq equations with fractional Laplacian. For $\alpha >\frac{5}{4}$ and $\beta >\frac{5}{8}$, by adopting the $\mathrm{\ell }$-trajectory method introduced in \cite{MP2002}, we demonstrate the existence of both a finite-dimensional pullback attractor and a pullback exponential attractor for the process $\{L(t,\tau ){\}}_{t\ge \tau }$ that corresponds to \cref{eq0101} in the phase space $X_{\mathrm{\ell }}$. Furthermore, we establish the existence of a pullback exponential attractor for the process $\{U(t,\tau ){\}}_{t\ge \tau }$ within the original phase space $H_{\sigma }^{\alpha }(\mathrm{\Omega })\times H^{\beta }(\mathrm{\Omega })$.