I am reading a paper and I really don't understand the following how hard I think about it.
For a $\nu \in \mathbb{R}_+$, let $v_\nu$ be the solution of the following incompressible Navier-Stokes system
\begin{align*} (NS_{\nu}) \begin{cases} \partial_t v_\nu + v_\nu \cdot \nabla v_\nu - &\nu \Delta v_\nu = -\nabla p_\nu\\ &\nabla \cdot v_\nu = 0\\ &v_\nu|_{t=0} = v^0 \end{cases}. \end{align*}
Also, let $v$ be the solution of the following incompressible Euler system.
\begin{align*} (E) \begin{cases} \partial_t v + &v \cdot \nabla v = -\nabla p_\nu\\ &\nabla \cdot v = 0\\ &v|_{t=0} = v^0 \end{cases}. \end{align*}
If we define $w_\nu = v_\nu - v$, by difference between $(NS_\nu)$ and $(E)$, we get $$ \partial_t w_\nu + v_\nu\cdot \nabla w_\nu = -\nabla \bar{p}_\nu + \nu \Delta v_\nu - w_v \cdot \nabla v$$
Then, can we get $\Vert \nabla v_\nu(t) \Vert_{L^2} \le \Vert w_\nu (t)\Vert_{L^2}$ from the above?
It seems that the author says it is a result of classical harmonic analysis, but I am not sure which part of harmonic analysis he is referring to. It seems similar to Poincare inequality, but it is not. Could you help me understand this or at least tell me which part of harmonic analysis it is about? Thank you in advance.