We study the Navier-Stokes-Darcy-Boussinesq system that models the thermal convection of a fluid overlying a saturated porous medium in a general decomposed domain. In both two and three spatial dimensions, we first prove the existence of global weak solutions to the initial boundary value problem subject to the Lions and Beavers-Joseph-Saffman-Jones interface conditions. The proof is based on a proper time-implicit discretization scheme combined with the Leray-Schauder principle and compactness arguments. Next, we establish a weak-strong uniqueness result such that a weak solution coincides with a strong solution emanating from the same initial data as long as the latter exists.

The paper investigates the existence of strong global and exponential attractors and their continuity for a class of semilinear wave equations with fractional damping in $\Omega\subset \mathbb R^3$: $$ u_{tt}-\Delta u+\gamma (-\Delta)^\theta u_t +f(u)=g(x), $$ with dissipative index $\theta \in (0, {\frac 12 } )$, It shows that when the growth exponent $p$ of the nonlinearity $f(u)$ is up to the range: $1\leq p < p_\theta:=3+4\theta$, (i) the weak solutions of the IBVP of the equation are of optimal higher regularity as $t > 0$, which leads to the fact that the weak solutions are exactly the strong ones; (ii) the related solution semigroup $S^\theta(t)$ acting on natural energy space ${\mathcal H}$ has a strong $({\mathcal H},{\mathcal H}_1)$-global attractor ${\mathscr A}_\theta$ and a strong $({\mathcal H},{\mathcal H}_1)$-exponential attractor ${\mathscr E}_\theta$, which are upper semicontinuous and continuous on $\theta$ at each point $\theta_0\in (0, {\frac 12 } )$ in the topology of strong space ${\mathcal H}_1$, respectively. The novelty of the paper is that it provides a new approach to establish the existence of the strong solutions while the initial data is in weaker space, and to study the existence of the strong global and exponential attractors and their continuity on the dissipative index $\theta$ in the topology of the strong space. These results improve and deepen those in recent literatures [35, 36, 37, 38].

In this paper, we consider a system related to Maxwell's equations, which has different structure with general elliptic systems. The existence of the weak solution is obtained in Reifenberg domain. By assuming that the coefficient matrix belongs to BMO with small seminorm, we establish the $W^{1,p}$ regularity in Reifenberg domain. The main analytical tools are the modified Vitali covering lemma, the maximal function technique and the compactness method.