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