  The Dean-Kawasaki (DK) equation is a stochastic partial differential equation
(SPDE) for the global density $\rho$ of a gas of $N$ over-damped Brownian
particles. In the thermodynamic limit $N\rightarrow \infty$ with weak pairwise
interactions, the expectation ${\mathbb E}[\rho]$ converges in distribution to
the solution of a McKean-Vlasov (MV) equation. In this paper we derive a
generalized DK equation for an interacting Brownian gas in a partially
absorbing one-dimensional medium. In the case of the half-line with a totally
reflecting boundary at $x=0$, the generalized DK equation is an SPDE for the
joint global density
$\rho(x,\ell,t)=N^{-1}\sum_{j=1}^N\delta(x-X_j(t))\delta(\ell-L_j(t))$, where
$X_j(t)$ and $L_j(t)$ denote the position and local time of the $j$th particle,
respectively. Assuming the DK equation has a well-defined mean field limit, we
derive the MV equation on the half-line with a reflecting boundary, and analyze
stationary solutions for a Curie-Weiss (quadratic) interaction potential. We
then use an encounter-based approach to develop the analogous theory for a
partially absorbing boundary at $x=0$. Each particle is independently absorbed
when its local time $L_j(t)$ exceeds a random threshold $\widehat{\ell}_j$ with
probability distribution $\Psi(\ell)=\P[\widehat{\ell}_j>\ell]$. The joint
global density is now summed over the set of particles that have not yet been
absorbed, and expectations are taken with respect to the Gaussian noise and the
random thresholds $\widehat{\ell}_j$. Extensions to finite intervals and
partially absorbing traps are also considered.
