For some numerical analysis of a fluid, I am wondering if there is **any** inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In particular, I am seeking one that is useful for the form:
$$(D(x),D(v))\_{L^2} + (\nabla\cdot{x}, \nabla\cdot{v})\_{L^2} + (\nabla\cdot{v}, \nabla\cdot{v})\_{L^2}\ge C\|\nabla x\|^2\_{L^2} + K\|\nabla v\|^2\_{L^2}$$
for constants $C > 0$ and $K \ge 0$, where strain-rate tensor $D(v)\equiv 
\frac12(\nabla v + (\nabla v)^T)$, $t$ represents time, and velocity $v = \frac{d}{dt}x$ for displacement $x$.


I am trying to see if I can show that the left hand side is coercive in terms of $\nabla x$. i.e. it is okay if $K$ may be zero.


$x$ and $v$ are in the Sobolev space $V=\{\varphi \in (H^1(\Omega))^d \;|\; \varphi = 0\textrm{ on } \partial\Omega\}$ where $\Omega$ is a $d-$dimensional region of the fluid flow, for $d=2,3$.


**P.S.** If the inequality were to be reversed, I would have used the fact that $\|\nabla\cdot{v}\|\_{L^2} \le \sqrt{d}\|\nabla{v}\|\_{L^2}$ for a $d$-dimensional domain; and also applied Cauchy-Schwarz inequality, then Young's inequality, and Korn's inequality.


[This answer](https://math.stackexchange.com/a/1448121/10803) on Math SE mentions Kantorovich inequalities, but there is no reference and my searches so far have been futile.



Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?
I don't think what you want is possible. 


In phase space $x$ and $v$ are basically quantities that can be independently prescribed. The first terms of your left hand side is linear in $v$. So fixing $x$, $v$ arbitrary, for $\lambda$ of the correct sign and $|\lambda|$ sufficiently small, you will have that the left hand side is negative if you replace $v$ by $\lambda v$. A negative quantity can never be coercive!


