Fokker Planck Equation with strongly convex potential function

Question

Let $\rho_t$ be the (weak) solution of Fokker-Planck equation $$ \partial_t\rho_t = \nabla\cdot(\rho_t\nabla(\log\rho_t + V)) $$ with initial condition $\rho_0$ and no-flux boundary condition $\rho_t(\nabla(\log\rho_t + V))\cdot n = 0$ on the boundary $\partial\Omega$ of a compact set $\Omega\subset \mathbb{R}^d$. Can we prove $$ \int_0^t\!\!\int_{\Omega} \Big|\Big|\nabla\log\frac{\rho_t(x)}{\rho_0(x)}\Big|\Big|^2 d\rho_0(x)dt \leq Ct^{\alpha},\qquad\forall\,t\in[0, \tau] $$ for some constant $C = C(\Omega, \rho_0)$, $\alpha > 0$ (I expect $\alpha=1$), and $\tau>0$ (may small enough)? Here, we can assume $\partial\Omega$ is smooth, $V$ is smooth and strongly convex.