Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

Finite-time singularities in the dynamical evolution of contact lines

We study finite-time singularities in the linear advection-diffusion equation with a variable speed on a semi-infinite line. The variable speed is determined by an additional condition at the boundary, which models the dynamics of a contact line of a hydrodynamic flow at a 180 contact angle. Using apriori energy estimates, we derive conditions on variable speed that guarantee that a sufficiently smooth solution of the linear advection--diffusion equation blows up in a finite time. Using the class of self-similar solutions to the linear advection-diffusion equation, we find the blow-up rate of singularity formation. This blow-up rate does not agree with previous numerical simulations of the model problem.Comment: 9 pages, 2 figure

The thinning of the liquid layer over a probe in two-phase flow

The draining of the thin water film that is formed between a two dimensional, infinite, initially flat oil-water interface and a smooth, symmetric probe, as the interface is advected by a steady and uniform flow parallel to the probe axis, is modelled using classical fluid dynamics. The governing equations are nondimensionalised using values appropriate to the oil extraction industry. The bulk flow is driven by inertia and, in some extremes, surface tension while the viscous effects are initially confined to thin boundary layers on the probe and the interface. The flow in the thin water film is dominated by surface tension, and passes through a series of asymptotic regimes in which inertial forces are gradually overtaken by viscous forces. For each of these regimes, and for those concerning the earlier stages of approach, possible solution strategies are discussed and relevant literature reviewed. Consideration is given to the drainage mechanism around a probe which protrudes a fixed specified distance into the oil. A lubrication analysis of the thin water film may be matched into a capillary-static solution for the outer geometry using a slender transition region if, and only if, the pressure gradient in the film is negative as it meets the static meniscus. The remarkable result is that, in practice, there is a race between rupture in the transition region and rupture at the tip. The analysis is applicable to the case of a very slow far field flow and offers significant insight into the non-static case. Finally, a similar approach is applied to study the motion of the thin water film in the fully inviscid approximation, with surface tension and a density contrast between the fluids

On the dynamics of random neuronal networks

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the limit in distribution of the sample path of the state of a neuron of the network when its size gets large. The invariant distributions of this limiting stochastic process are analyzed as well as their stability properties. We show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.Comment: 37 pages, 3 figure