just lubrication forces between particles but also long range forces, wall effects and more. In the case of inter-particle HI, the diagonal blocks $\bm{\Gamma}_{ii}$ each represent the force exerted on the fluid due to the motion of particle $i$, which is simply the sum of all the pairwise HI from the perspective of particle $i$. The off-diagonal blocks $\bm{\Gamma}_{ij}$ represent the force on particle $i$ due to the motion of particle $j$.

The stationary equations for the equilibrium density $\varrho(\vec{r})$ and equilibrium flux $\vec{a}(\vec{r})$ are given by

$$\begin{aligned}
&\qquad\qquad\qquad\qquad\nabla_{\vec{r}}\cdot \vec{a}(\vec{r}) = 0,\label{eq:div_a_0}\\
&\vec{a}(\vec{r}) + \boldsymbol{H}[\vec{a},\varrho](\vec{r})+\frac{\varrho(\vec{r},t)}{k_BT}\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r})
=0.\label{eq:eqn_for_a_in_equilibrium}
 \end{aligned}$$

Note that given a finite flux vector $\vec{a}$ solving -, it is not obvious that $\varrho$ is necessarily a minimiser of the free energy $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$ (where $\mu_{c}$ is the chemical potential of the species). However, for the particular choice $\vec{a}\equiv \vec{0}$ (which is a natural and physically realistic solution), $\varrho$ is necessarily a minimiser of $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$, and we will show that under reasonable assumptions these are indeed the only fixed points of the system.

Previous well-posedness studies of similar nonlinear, nonlocal PDEs focused on periodic boundary conditions; see, e.g., [@chazelle2017well; @greg_mckean_vlasov_torus]. In contrast, we are interested in the well-posedness of , - subject 