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In this paper, we consider an interface problem between a fluid flow, governed by Navier-Stokes equations, and a flow in a porous medium governed by the Darcy equation. We focus on the Navier-Stokes-Darcy system with the interface condition more physically correct which is rigorously derived from the balance of forces (see [V. Girault, B. Rivière (2009), [22] ]). Based on the local well-posedness and some key a priori bounds established, we obtain the global existence and uniqueness of the strong solution to the coupled problem. Subsequently, we also give the exponential stability of the strong solution which, to our knowledge, is the first description of the large-time behavior of the solution to the Navier-Stokes-Darcy system. [ABSTRACT FROM AUTHOR]

This paper is concentrated upon the qualitative analysis of two specific cases of the compressible Euler equations with linear damping: self-balanced non-isentropic system and externally driven isentropic system. We consider initial-boundary value problems of the models on bounded domains in R d. Both systems are supplemented with the no-normal-flow boundary condition. Under smallness assumptions on the initial perturbation and/or external force, it is shown that non-trivial steady states associated with the initial-boundary value problems are asymptotically stable in the long run. [ABSTRACT FROM AUTHOR]