Abstract The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the $$L_p$$ L p -setting. More precisely, the existence and uniqueness of a local, strong $$L_p$$ L p -solution to the general system is proved and it is shown that the director d satisfies $$|d|_2\equiv 1$$ | d | 2 ≡ 1 provided this holds for its initial data $$d_0$$ d 0 . In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

Abstract This paper analyzes weak solutions of the quantum hydrodynamics (QHD) system with a collisional term posed on the one-dimensional torus. The main goal of our analysis is to rigorously prove the time-relaxation limit towards solutions to the quantum drift-diffusion (QDD) equation. The existence of global in-time, finite energy weak solutions can be proved by straightforwardly exploiting the polar factorization and wave function lifting tools previously developed by the authors. However, the sole energy bounds are not sufficient to show compactness and then pass to the limit. For this reason, we consider a class of more regular weak solutions (termed GCP solutions), determined by the finiteness of a functional involving the chemical potential associated with the system. For solutions in this class and bounded away from vacuum, we prove the time-relaxation limit and provide an explicit convergence rate. Our analysis exploits compactness tools and does not require the existence (and smoothness) of solutions to the limiting equations or the well-preparedness of the initial data. As a by-product of our analysis, we also establish the existence of global in time $$H^2$$ H 2 solutions to a nonlinear Schrödinger–Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.

Abstract Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. Using the Itô–Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.