uniqueness, positivity and regularity of the weak solution to when $\vec{a}(\vec{r},t)$ is given by the integral equation

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

$$\begin{aligned}
\boldsymbol{H}[\vec{a},\varrho](\vec{r},t):=
\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}')\boldsymbol{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r}',t),\label{eq:eqn_for_H}\end{aligned}$$

$$\begin{aligned}
&\frac{\varrho(\vec{r},t)}{k_BT}\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r},t) := [\nabla_{\vec{r}}+\tfrac{1}{k_B\mathrm{T}}\Big(\nabla_{\vec{r}}V_1(\vec{r},t)\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\int_U\mathrm{d}\vec{r}'\varrho(\vec{r}',t)g(\vec{r},\vec{r}')\nabla_{\vec{r}}V_{2}(\vec{r},\vec{r}')\Big)  ]\varrho(\vec{r},t),\label{eq:eqn_for_J}\end{aligned}$$

where to ease notation we have suppressed $[\varrho]$ in the argument of $\vec{a}$ and $\mathcal{F}$ is the free energy functional which will be defined in Section \[subsec:free\_energy\_framework\]. The functions $V_1$ and $V_2$ are the external and (two body) interparticle potentials respectively. Additionally, the non-constant diffusion tensor $$\begin{aligned}
\label{eq:def_diffusion_tensor}
\boldsymbol{D}(\vec{r},[\varrho],t):=\frac{k_{\text{B} }\mathrm{T}}{m \gamma}\Big[\boldsymbol{1}+\int\mathrm{d}\vec{r}'g(\vec{r},\vec{r}')\varrho(\vec{r}',t)\boldsymbol{Z}_1(\vec{r},\vec{r}')\Big]^{-1}\end{aligned}$$ will be considered; this is interesting from a physical point of view. It has been previously shown (see [@goddard2012overdamped]) that for $\boldsymbol{Z}_1$ being positive definite, $\boldsymbol{D}$ is also positive definite and therefore has positive, finite eigenvalues. The term $g(\vec{r},\vec{r}')$ (regarded as known) is the correlation function defined by the two-body density $\varrho^{(2)}(\vec{r},\vec{r}',t) =g(\vec{r},\vec{r}')\varrho(\vec{r},t)\varrho(\vec{r}',t)$ and the operator $\boldsymbol{H}[\cdot]$ describes terms corresponding to HI.

We note that if $\boldsymbol{D}$ were positive semidefinite, a zero eigenvalue of $\boldsymbol{D}$ is permitted, which physically-speaking would amount to the colloidal system possessing a zero diffusion rate in some subset of $U$ with nonzero measure. Such systems are interesting (for example, in many biological systems the physical domain $U$ could be a substrate including 