# Robustness of 3D Navier-Stokes System with Increasing Damping
## 4 Proof  
### Estimation of Solutions  
**Lemma 2**.: _Let \(3<\gamma\leq 5\), \(u(\varsigma)\in\mathcal{V}\cap(L^{\gamma+1})^{3}\), and \(t\in\mathbb{R}\) is arbitrary. Then, there exist \(T_{0}>0\) and \(\rho_{1}(t)>0\) satisfying_  
\[|u(t)|^{2}\leq\rho_{1}^{2}(t),\ \varsigma<-T_{0}.\]  
Proof.: Multiplying (3) by \(e^{\delta t}u\), we can get  
\[\frac{d}{dt}(e^{\delta t}|u(t)|^{2})+K_{1}e^{\delta t}(\|u\|^{2}+\|u\|_{\gamma+ 1}^{\gamma+1})\leq\frac{\varepsilon^{2}}{\
u\lambda_{1}}e^{\delta t}|g|^{2}, \tag{4}\]  
where \(K_{1}=\min\{v-\frac{\delta}{\lambda_{1}},2\beta\}\), then integrating it over \([\varsigma,t]\), we also get  
\[e^{\delta t}|u(t)|^{2}+K_{1}\int_{\varsigma}^{t}e^{\delta s}(\|u(s)\|^{2}+\|u( s)\|_{\gamma+1}^{\gamma+1})ds\leq\frac{\varepsilon^{2}}{\
u\lambda_{1}}\int_{- \infty}^{t}e^{\delta s}|g|^{2}ds+e^{\delta\varsigma}|u(\varsigma)|^{2}. \tag{5}\]  
Finally, there holds  
\[|u(t)|^{2}\leq e^{\delta(\varsigma-t)}|u(\varsigma)|^{2}+\frac{\varepsilon^{2 }}{\
u\lambda_{1}}e^{-\delta t}\int_{-\infty}^{t}e^{\delta s}|g|^{2}ds,\]  
which means there is a positive constant \(T_{0}\) satisfying  
\[|u(t)|^{2}\leq\frac{2\varepsilon^{2}}{\
u\lambda_{1}}e^{-\delta t}\int_{- \infty}^{t}e^{\delta s}|g|^{2}ds\equiv\rho_{1}^{2}(t),\ \forall\ \varsigma\leq-T_{0}. \tag{6}\]  
**Lemma 3**.: _Let \(3<\gamma\leq 5\), \(u(\xi)\in\mathcal{V}\cap(L^{\gamma+1})^{3}\), and \(t\in\mathbb{R}\) is arbitrary. Then, there are positive constants \(T_{1}\) and \(\rho_{2}(t)\) satisfying_  
\[\|u(t)\|^{2}+\|u(t)\|_{\gamma+1}^{\gamma+1}\leq\rho_{2}^{2}(t),\ \xi\leq-T_{1}.\]  
Proof.: Multiplying (3) by \(u_{t}\) and \(Au\), respectively, we have  
\[2|u_{t}|^{2}+v\frac{d}{dt}\|u\|^{2}+\frac{2\beta}{\gamma+1}\frac {d}{dt}\|u\|_{\gamma+1}^{\gamma+1}\] \[\leq 2|((u\cdot\
abla)u,u_{t})|+2|(u_{t},\epsilon g)|\leq|u_{t}|^ {2}+2|u\cdot\
abla u|^{2}+2|\epsilon g|^{2} \tag{7}\]  
and  
\[\frac{d}{dt}\|u\|^{2}+2v|Au|^{2}+2\beta||u|^{\gamma-1}\
abla u|^{ 2}+\frac{8\beta(\gamma-1)}{(\gamma+1)^{2}}|\
abla|u|^{\frac{\gamma+1}{2}}|^{2}\] \[\leq 2|((u\cdot\
abla)u,Au)|+2|(Au,\epsilon g)|\leq\frac{\
u}{2}| Au|^{2}+\frac{4}{\
u}|u\cdot\
abla u|^{2}+\frac{4}{\
u}|\epsilon g|^{2}. \tag{8}\]  
Using the estimates in [20] that  
\[C|u\cdot\
abla u|^{2}\leq\beta||u|^{\gamma-1}\
abla u|^{2}+\frac{ \
u}{2}|Au|^{2}+C\|u(t)\|_{\gamma+1}^{\gamma+1},\]  
and adding (7) and (8) together, we get  
\[(\
u+1)\frac{d}{dt}\|u\|^{2}+\frac{2\beta}{\gamma+1}\frac{d}{dt} \|u(t)\|_{\gamma+1}^{\gamma+1}+2\
u|Au|^{2}+2\beta||u|^{\gamma-1}\
abla u|^{2} +|u_{t}|^{2}\] \[\leq\beta||u|^{\gamma-1}\
abla u|^{2}+\frac{\
u}{2}|Au|^{2}+C\|u (t)\|_{\gamma+1}^{\gamma+1}+(\frac{4}{\
u}+2)\epsilon^{2}|g|^{2},\]  
and  
\[\frac{d}{dt}(\epsilon^{\beta t}(\|u\|^{2}+\|u\|_{\gamma+1}^{ \gamma+1}))+C\epsilon^{\beta t}(|Au|^{2}+||u|^{\gamma-1}\
abla u|^{2})\] \[\leq C\epsilon^{\beta t}(\|u(t)\|^{2}+\|u(t)\|_{\gamma+1}^{ \gamma+1})+C\epsilon^{2}e^{\beta t}|g|^{2}.\]  
By integration of the inequality and conclusion in Lemma 2, we conclude that  
\[\epsilon^{\beta t}(\|u(t)\|^{2}+\|u(t)\|_{\gamma+1}^{\gamma+1})\] \[\leq e^{\beta\xi}(\|u(\xi)\|^{2}+\|u(\xi)\|_{\gamma+1}^{\gamma+1} )+C\int_{\xi}^{t}\epsilon^{\beta s}(\|u(s)\|^{2}+\|u(s)\|_{\gamma+1}^{\gamma+1 })ds+C\epsilon^{2}\int_{-\infty}^{t}e^{\beta s}|g|^{2}ds\] \[\leq C\epsilon^{\beta\xi}(\|u(\xi)\|^{2}+\|u(\xi)\|_{\gamma+1}^{ \gamma+1})+C_{1}\epsilon^{2}\int_{-\infty}^{t}e^{\beta s}|g|^{2}ds+C_{2},\]  
and there are positive constants \(T_{1}\) and \(\rho_{2}(t)\) satisfying  
\[\|u(t)\|^{2}+\|u(t)\|_{\gamma+1}^{\gamma+1}\leq\rho_{2}^{2}(t),\ \forall\ \xi\leq-T_{1}.\]  
**Lemma 4**.: _Let \(3<\gamma\leq 5\), \(u(\xi)\in\mathcal{V}\cap(L^{\gamma+1})^{3}\), and \(t\in\mathbb{R}\) is arbitrary. Then, there are positive constants \(T_{2}\) and \(\rho_{3}(t)\) satisfying_  
\[|Au(t)|^{2}\leq\rho_{3}^{2}(t),\ \xi\leq-T_{2}.\]  
The proof can be referred to in [19].