Interfacial flows close to a moving contact line are inherently multi-scale. The shape of the interface and the flow at meso-and macroscopic scales inherit an apparent interface slope and a regularization length, both called after Voinov, from the dynamical processes at work at the microscopic level. Here, we solve this inner problem in the case of a volatile fluid at equilibrium with its vapor. The evaporative/condensation flux is then controlled by the dependence of the saturation temperature on interface curvature -the so-called Kelvin effect. We derive the dependencies of the Voinov angle and of the Voinov length as functions of the substrate temperature. The relevance of the predictions for experimental problems is finally discussed.The dynamics of a macroscopic solid plunging in a liquid bath [1,2] or withdrawn from it [3-5] depends sensitively on its wetting properties i.e. on the intermolecular interactions at the nanoscopic scale. The motion of the contact line separating wet from dry regions is therefore an inherently multi-scale problem. Amongst the important consequences of the coupling between inner and outer scales (Fig. 1), the speed at which a contact line can recede over a flat solid surface cannot exceed a critical value, associated to a dynamical wetting transition which leads to the formation of a dewetting ridge [6,7], of a V-shaped dewetting corner [1,[8][9][10] or to the entrainment of films [2, 10, 11] (see [12,13] for detailed reviews). In many applications, such as coating, imbibition of powders, immersion lithography or boiling-free heating, these entrainment phenomena are crucial limiting factors for industrial processes. Fig. 1 shows schematically the structure of the flow close to a moving contact line. Even for an infinitesimal velocity U , there exists a range of mesoscopic scales -roughly six decades -separating the microscopic scale from the macroscopic length L, in which the diverging viscous stress is balanced by a gradient of capillary pressure. This balance can be made quantitative in the lubrication approximation, for which the angles are assumed small, and which gives a third order differential equation for the interface profile h(x):where η is the liquid dynamic viscosity and γ the surface tension; U is positive for an advancing contact line. This equation has an exact solution [14] which reduces to the asymptotic form proposed by Voinov [15] far from the contact line, but for x ≪ L:θ V is by definition the apparent contact angle in the static case (U = 0), which can be different from the Young angle θ Y due to out of equilibrium processes taking place at a microscopic scale. The Voinov length ℓ V is also a quantity defined in the mesoscopic range of scales but inherited from the inner region, where the problem is regularized. The mesoscopic solution (2) must also be matched at the macroscopic scale L to an outer solution where viscosity can usually be neglected. Fig. 1 features the case of a spreading drop or of a growing bubble but the outer matching problem has been s...

International audienceWe explain and prove the statement from the title. This allows us to formulate a new type of gradient inequality and to obtain a new stabilization result for gradient-like ordinary differential equations

In this paper we show boundedness of vector-valued Bergman projections on simple connected domains. With this result we show R-sectoriality of the derivative on the Bergman space on C + and maximal L p -regularity for an integrodifferential equation with a kernel in the Bergman space.2000 Mathematics subject classification: 47D06, 32A36.

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The regularized analytical continuation method (RAC) counts among the most powerful modern method for determination of resonance energies and resonance widths of low energy atomic and molecular systems including large systems of biological importance. By this method the resonance state is transformed into bound state by addition of an attractive potential. In this paper the choice of the perturbation potential is discussed and the performance of the method studied using model potential. It is shown that for correct applications of the RAC method the knowledge of the asymptotic behavior of the basic ingredient of the method, the coupling function λ(κ), is important. In this paper we find the correct asymptotic form of λ(κ) for broad class of realistic potentials and for nonzero values of angular momentum. New RAC formula convenient for singular perturbation potentials is proposed.