Maximal monotonicity and contraction semigroup for the quantum drift-diffusion (Derrida-Lebowitz-Speer-Spohn) equation

Abstract

We study the quantum drift-diffusion, or Derrida-Lebowitz-Speer-Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt\varrho$. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^2(\Omega)$, 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 $L^2$ 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 $\sqrt u\in L^2_{\rm loc}(H^2)$ 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 B\'enilan integral solutions and through one-sided weak formulations - prove the maximality of the operator also in the $H^2$-$H^{-2}$ duality and, in dimension $d\le3$, 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 $\int_\Omega(\Delta u)^2/u\,\mathrm{d} x$ is finite exactly when $\sqrt u\in H^2(\Omega)$, and it then controls the full Hessian of $\sqrt u$, in every dimension.