# Robustness of 3D Navier-Stokes System with Increasing Damping
## 1 Introduction  
The Navier-Stokes system depicts the conservation law about mass and momentum of fluid, reflects the basic relationship among gravity, pressure and other factors inside the viscous fluid, and is widely applied in many fields, such as materials, weather forecast and so on. There has been a good deal of interesting research results on Navier-Stokes equations, such as well-posedness of system solution, asymptotical behaviour, existence and dimension evaluation of several attractors, determination, invariant measures and so on, which can be referred to [1; 2; 3; 4; 5; 6; 7; 8; 9]. Given that the 3D NS system is extremely complex, especially in the case of solution well-posedness, the 3D NS system with damping becomes one of the hot research topics. About the increasing damping form \(|u|^{\gamma-1}u\), it is necessary for us to study the porous media. Porous media is very common in our lives, such as coal, animal fur, aluminium foams, ceramics and so on. Further, the applications of porous media involve many aspects of our lives--heat transfer device design, sound testing in medicine, food drying, underground water flow, ice melting, sound propagation in building materials, noise reduction in automotive design, heat retention in materials, etc. The porosity is the most representative concept in studying porous media, which is the ratio of fluid volume to the total, a nonnegative number less than or equal to 1, where the fluid is the compound of air and some liquids. From the Darcy law, we know, in the flow within porous media, the flow speed \(u\) and the pressure \(p\) satisfy  
\[\mu u=-k\
abla p,\]  
where the viscosity coefficient \(\mu\) and permeability coefficient \(k\) are both positive. Further, the flow in saturated porous media can be described as  
\[\mu u=-k\
abla p+\rho f,\]  
where \(\rho\) denotes the fluid density in porous media and \(f\) is the force. When the flow speed in porous media becomes large to a certain extent, to obtain an accurate description for porous media, the left term \(\mu u\) will be replaced by a new one, such as \(au+bu^{2}+cu^{3}\)Therefore, the increasing damping form \(|u|^{\gamma-1}u\) appears naturally in the fluid system within the porous media, some detailed conclusions can be referred to [10]. Further, we can find some meaningful results on models with the increasing damping for non-Newtonian fluid in [11; 12; 13], and also the simulations on a system with the damping form \(|u|^{\gamma-1}u\) is one of the future research directions, where the ideas in [14; 15] can be used for reference. In this paper, we investigate the following three-dimensional incompressible NS system with the increasing damping \(|u|^{\gamma-1}u\)  
\[\left\{ \begin{array}{l}u_{t}-\
u\Delta u+(u\cdot\
abla)u+\beta|u|^{\gamma-1}u +\
abla p=\varepsilon g(x,t),\ \ (x,t)\in\Omega\times(\varepsilon,+\infty),\\ divu=0,\ \ (x,t)\in\Omega\times(\varepsilon,+\infty),\\ u(x,t)|\omega=0,\ t\in(\varepsilon,+\infty),\\ u(x,\varepsilon)=u(\varepsilon),\ x\in\Omega,\end{array} \right. \tag{1}\]  
where \(\Omega\subset\mathbb{R}^{3}\) is a bounded domain with a smooth boundary, \(u=u(x,t)\) is the unknown velocity field, \(p\) is the pressure, \(\
u>0\) is the viscosity coefficient, \(\varepsilon g(x,t)\) is the external force and \(\varepsilon>0\), \(\beta>0\), \(\gamma\geq 1\). Meaningful results on system (1) can be found in [16; 17; 18; 19; 20]. Where the well-posedness of the strong solution was shown in [16; 19], based on which three types of attractors were given in [17; 18; 20]. In practice, due to measurement errors and circumstances, the parameters of the system are inevitably perturbed. In this case, the study of robustness is particularly important, and robustness has become one of the important indicators of control system design. For the conclusion of studying the robustness through upper semicontinuity, we could refer to [21; 22; 23; 24; 25].