# Robustness of 3D Navier-Stokes System with Increasing Damping
## 2 Preliminaries  
### Some Definitions and Theorems  
Consider a Banach space \(\mathcal{X}\) with metric \(d_{\mathcal{X}}(\cdot,\cdot)\) and norm \(\|\cdot\|_{\mathcal{X}}\). \(\{S(t,\xi)\}_{t\geq\xi}\) is a process in \(\mathcal{X}\), its definition and properties can be referred to in [26]. **Definition 1**. : _Suppose that \(\tilde{\mathcal{A}}(t)\) is a family of compact subsets \(\{\tilde{A}(t)\}_{t\in\mathbb{R}}\), and satisfies_  
(i) _Characteristic of invariance: \(S(t,\xi)\tilde{A}(\xi)=\tilde{A}(t),\ \forall t\geq\varsigma\)._  
(ii) _Characteristic of pullback attraction: for any subset \(\tilde{D}\) bounded in \(\mathcal{X}\), there holds_  
\[\lim_{\varsigma\to+\infty}dist_{\mathcal{X}}(S(t,t-\varsigma)\tilde{D},\tilde{ A}(t))=0,\]  
_then we call \(\tilde{\mathcal{A}}(t)=\{\tilde{A}(t)\}_{t\in\mathbb{R}}\) the pullback attractor._  
**Definition 2**. : _For arbitrary subset \(\tilde{D}\) bounded in \(\mathcal{X}\), if there always exists \(\tilde{T}(t,\tilde{D})>0\) satisfying_  
\[S(t,t-\varsigma)\tilde{D}\subset\tilde{D}(t),\ \varsigma\geq\tilde{T}(t,\tilde{D} ),\ t\in\mathbb{R},\]  
_then we call the subset family \(\tilde{\mathcal{D}}=\{\tilde{D}(t)\}_{t\in\mathbb{R}}\) the pullback absorbing set of \(\{S(t,t-\varsigma)\}\)._  
**Definition 3**. : _For arbitrary \(t\in\mathbb{R}\), let \(\tilde{\mathcal{D}}=\{\tilde{D}(t)\}_{t\in\mathbb{R}}\) be a subset family in \(\mathcal{X}\). For arbitrary sequence \(\{\varsigma_{n}\}\) with \(\varsigma_{n}\to+\infty\)\((n\to+\infty)\) and \(\xi_{n}\in\tilde{D}(t-\varsigma_{n})\), if \(\{S(t,t-\varsigma_{n})\xi_{n}\}\) possesses the relative compactness in space \(\mathcal{X}\), then we call \(\{S(t,t-\varsigma)\}\) pullback \(\tilde{\mathcal{D}}-\)asymptotically compact in space \(\mathcal{X}\)._  
**Theorem 1**. : _Suppose that \(\tilde{\mathcal{D}}=\{\tilde{D}(t)\}_{t\in\mathbb{R}}\) is the pullback-absorbing set for \(\{S(t,t-\varsigma)\}\), which is continuous and possesses the pullback \(\tilde{\mathcal{D}}-\)asymptotential compactness in \(\mathcal{X}\) topology. Then the subset family \(\tilde{\mathcal{A}}(t)=\{\tilde{A}(t)\}_{t\in\mathbb{R}}\), where_  
\[\tilde{A}(t)=\Lambda(\tilde{\mathcal{D}},t)=\underset{s\geq 0}{\cap}\overline{ \underset{\varsigma\geq s}{\cup}S(t,t-\varsigma)\tilde{D}(t-\varsigma)}^{ \mathcal{X}},\]  
_is the pullback attractor to \(\{S(t,t-\varsigma)\}\) in space \(\mathcal{X}\)._  
**Definition 4**. : _Let \(\varepsilon\in(\varepsilon_{0}-h,\varepsilon_{0}+h)\), \(\{\tilde{D}_{\varepsilon}(t)\}_{t\in\mathbb{R}}\subset\mathcal{X}\) denotes a subset family in space \(\mathcal{X}\), and function \(\Phi(t,\cdot\cdot,\cdot,\cdot)\)\((t\in\mathbb{R})\) is defined on_  
\[(\varepsilon_{0}-h,\varepsilon_{0}+h)\times(\varepsilon_{0}-h,\varepsilon_{0 }+h)\times\underset{\varepsilon\in(\varepsilon_{0}-h,\varepsilon_{0}+h)}{ \cup}\tilde{D}_{\varepsilon}(t)\times\underset{\varepsilon\in(\varepsilon_{0 }-h,\varepsilon_{0}+h)}{\cup}\tilde{D}_{\varepsilon}(t).\]  
_If for any sequences \(\{\varepsilon_{n}\}_{n\in\mathbb{N}}\subset(\varepsilon_{0}-h,\varepsilon_{0 }+h)\) and \(\{\xi_{n}\}_{n\in\mathbb{N}}\subset\tilde{D}_{\varepsilon_{n}}(t)\), there always exist subsequences \(\{\varepsilon_{n_{k}}\}_{k\in\mathbb{N}}\) and \(\{\tilde{\varepsilon}_{n_{k}}\}_{k\in\mathbb{N}}\) such that_  
\[\lim_{k\to\infty}\lim_{l\to\infty}\Phi(t,\varepsilon_{n_{k}},\varepsilon_{n_ {l}},\xi_{n_{k}},\xi_{n_{l}})=0,\]  
_then the function \(\Phi(t,\cdot\cdot,\cdot,\cdot)\) is said to be contractive._  
**Definition 5**. : _Let \(dist_{\mathcal{X}}(\cdot,\cdot)\) be the Hausdorff semidistance in \(\mathcal{X}\), if there holds_  
\[(\mathrm{H})\quad\lim_{\varepsilon\to\varepsilon_{0}}dist_{\mathcal{X}}(\tilde {A}_{\varepsilon}(t),\tilde{A}_{\varepsilon_{0}}(t))=0,\;\forall t\in\mathbb{R},\]  
_then it is said that \(\tilde{\mathcal{A}}_{\varepsilon}(t)\) is upper semi-continuous at \(\varepsilon_{0}\)._  
**Theorem 2** ([27]).