and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2_cosine.pdf "fig:"){width="\textwidth"}

Description of Main Results and Organisation of the Paper.
----------------------------------------------------------

### Main Results {#main-results .unnumbered}

The main results of this work are threefold.

1.  We establish existence and uniqueness of weak solutions to DDFTs including two-body HI governed by equations , - with no-flux boundary conditions.

2.  We derive *a priori* convergence estimates of the density $\varrho(\vec{r},t)$ to equilibrium in $L^2$ and relative entropy.

3.  We study the stability of equilibrium states and construct bifurcation diagrams for two numerical applications.

These results are of particular interest for physical applications of colloidal systems where conservation of mass is either a desirable or necessary property of the system. Additionally, the stability theorem contrasts with simpler linear stability analyses of similar systems of gradient flow structure with periodic boundary conditions [@martzel2001mean], [@greg_mckean_vlasov_torus] which may be tackled by means of Fourier analysis.

### Organisation of the Paper {#organisation-of-the-paper .unnumbered}

The paper is organised as follows: in Section \[sec:preliminaries\] 