In this paper, we show that the strong solution of the three-dimensional Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u\ (\alpha>0, \frac{7}{2}\leq \beta\leq 5)$ has global attractors in $V$ and $H^2(\Omega)$ when initial data $u_0\in V$, where $\Omega\subset \mathbb{R}^3$ is bounded.

This paper is concerned with a single-component model of heat and vapor (sweat) transport through three-dimensional porous textile materials with phase change, which is described by a nonlinear, degenerate, and strongly coupled parabolic system. An uncoupled (splitting) Galerkin method with semi-implicit Euler scheme in time direction is proposed for the system. In this method, a linearized scheme is applied for the approximation to Darcy's velocity simultaneously in the mass and energy equations, which leads to physical conservation of the method in the flow convection. The existence and uniqueness of solution of the finite element system is proved and the optimal error estimate in an energy norm is obtained. Numerical results are presented to confirm our theoretical analysis and are compared with experimental data.