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-Schwartz inequality, then Young's inequality, and Korn's inequality.

This answer on Math SE mentions Kantorovich inequalities, but there is no reference and my searches so far have been futile.