We study the quantum drift-diffusion, or Derrida-Lebowitz-Speer-Spohn (DLSS), equation for a nonnegative density ϱ on a bounded convex domain with Neumann boundary conditions, in the square-root variable u=ϱ. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in L2(Ω)
Nearby in the stack
, explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint. The generated semigroup, which contracts the Hellinger distance between the densities, thus yields a canonical solution - existing, unique, and stable for every nonnegative
L2
initial datum and in every space dimension - independent of any approximation scheme: it is in fact the unique contraction semigroup extending the classical evolutions that emanate from smooth, uniformly positive data. The implicit Euler scheme converges to it, and
u∈Lloc2(H2)
along the flow. When the datum belongs to the domain of the operator, the solution is strong and satisfies the equation pointwise, with no reaction term created on the vacuum
{u=0}
. We characterize the trajectories in several equivalent ways - as Bénilan integral solutions and through one-sided weak formulations - prove the maximality of the operator also in the
H2
-
H−2
duality and, in dimension
d≤3
, identify the flow with the weak solutions in the uniqueness class of Fischer. A second-order estimate of independent interest underlies the construction: on a convex domain with Neumann conditions the dissipation