I'm reading about gradient flows in Wasserstein space in this note.
Let $\mathcal{P}_2 (\mathbf{R}^d)$ be the set of probability measures with finite second moment. In [Ott01] the tangent space $T_\rho \mathcal{P}_2\left(\mathbf{R}^d\right)$ of $\rho \in \mathcal{P}_2 (\mathbf{R}^d)$ is defined by $$ s \in T_\rho \mathcal{P}_2\left(\mathbf{R}^d\right) \iff \color{blue}{s=-\nabla \cdot(\rho \nabla u)}, $$ and the inner product by $$ g_\rho(s, s)=\int_{\mathbf{R}^d}|\nabla u|^2 d \rho . $$ Hence for $s_1, s_2 \in T_\rho \mathcal{P}_2\left(\mathbf{R}^d\right)$ we have $$ g_\rho\left(s_1, s_2\right)=\frac{1}{4}\left[g_\rho\left(s_1+s_2\right)-g_\rho\left(s_1-s_2\right)\right]=\int_{\mathbf{R}^d} \nabla u_1 \cdot \nabla u_2 d \rho, $$ where $s_1=-\nabla \cdot\left(\rho \nabla u_1\right)$ and $s_2=-\nabla \cdot\left(\rho \nabla u_2\right)$.
My understanding I think the equation $s=-\nabla \cdot(\rho \nabla u)$ holds in distribution sense. However, I'm not familiar with PDEs and not sure what it means exactly. My guess is that $s:\mathbf R^d \to \mathbf R$ and $u:\mathbf R^d \to \mathbf R^d$ such that $$ \int_{\mathbf R^d} (s(x) + \langle \nabla \phi (x), u(x)) ) \mathrm d \rho(x)=0 \quad \forall \phi \in \mathcal C^\infty_c (\mathbf R^d). $$
Could you elaborate on the meaning of $s=-\nabla \cdot(\rho \nabla u)$?