Let $\mathbf H$ be a fixed Hilbert space. Let $\mathcal L(\mathbf H)$ be the collection of linear operators (that may not be bounded) on $H$. For every $A\in \mathcal L(\mathbf H)$, denote by $D(A)$ the domain of $A$. I am looking for the references on the following differential equation
$$\frac {{\rm d}A_t}{{\rm d}t} = LA_t + f(A_t)A_t,$$
where $L\in \mathcal L(\mathbf H)$ is some operator, $LA$ denotes the multiplication between operators, $f: \mathcal L(\mathbf H)\to\mathbb R$ (or $f: \mathcal L(\mathbf H)\to\mathbb C$) is some suitable functional, e.g. $f(A_t)=trace(A_t)$, $f(A_t)=trace(A_t^2)$, $f(A_t)=(trace(A_t))^2$, etc.
I am happy to know the existing literature on the study of wellposedness and solution properties of the above equation.
PS : Suggested by Willie Wong, we consider an example as below (which is the main motivation). Consider the partial differential equation on $\mathbb R_+\times \mathbb R^d \times\mathbb R^d$:
\begin{eqnarray*} \partial_t \rho(t,x,y) &=& -\mathrm{i}\Big(-\frac 12 (\Delta_{x}-\Delta_{y})\rho(t,x,y) +\big( V(x)-V(y)\big) \rho(t,x,y)\Big)\\ && - \frac{1}{2}\Big(K(x)- K(y) \Big)^2 \rho(t,x,y), \end{eqnarray*} where $V, K:\mathbb R^d\to\mathbb R$ are suitable functions. Here we pursue the solution $\rho$ in the trace class. When $K\equiv 0$, it is known that $\rho(t,x,y)=\varphi(t,x)\overline\varphi(t,y)$, where $\varphi$ solves some Schrödinger equation, see e.g. Energy estimation of density operator to von Neumann equation
For a general $K$, I wish to know how to study this wellposedness. A natural idea, as above, is to treat this PDE as a differential equation valued in operator space. Namely, $\rho_t:=\rho(t,\cdot)$ can be seen as an operator, and $V, K$ can be seen as multiplication operators, then
\begin{equation} \mathrm{d} \rho_t = -\mathrm{i}\left[-\frac 12 \Delta + V,\rho_t\right]\mathrm{d} t + \left(K\rho_tK - \frac{1}{2}K^2\rho_t-\frac{1}{2}\rho_tK^2 \right)\mathrm{d} t\quad (\ast) \end{equation}
Here $[\cdot,\cdot]$ denotes the commutator of two operators. Does $(\ast)$ admit a solution in the trace class?
