Monotone flows in N-dimensional partially saturated porous media: Lipschitz-continuity of the interface

Communicated by J. B. McLeod     

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flamesJustification mathématique d'une équation intégro-différentielle non linéaire pour un modèle de flamme sphérique

This Note is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [7]. The paper [7] derives, by means of a three-scale matched asymptotic expansion, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis number – i.e., the ratio between thermal and molecular diffusion – to be strictly less than unity. In this Note, we give the main ideas of a rigorous proof of the validity of this model, under the additional restriction that the Lewis number is close to 1. To cite this article: C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.     

A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions u^e{{u^\varepsilon}} to the singular perturbation problem and for u=limu^e{u=\lim{u^\varepsilon}} , assuming that both u^e{{u^\varepsilon}} and u were defined in an arbitrary domain D{\mathcal{D}} in \mathbbR^N+1{\mathbb{R}^{N+1}} . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while u^e{{u^\varepsilon}} are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u ₀ of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u ₀ in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport.     

粉尘空气混合物层流燃烧速度的实验测定

给出了计算粉尘层流火焰速度的直接方法,此法简便易行,在粉尘浓度较低时,计算精度较高。实验结果表明:粉尘浓度对火焰传播速度和燃烧速度有很大的影响,粉尘浓度过大时,粒子运动轨迹就难观测,用直接法计算层流火焰速度的误差增大;管径大小也对燃烧速度有很大影响;小管径中的所得值比大管径中的所得值约低８％。     

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [17]. The paper [17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number – i.e. the ratio between thermal and molecular diffusion – to be strictly less than unity. If is the inverse of the – reduced – activation energy, the idea underlying the construction of [17] is that (i) the time scale of the radius motion is ^-2, and that (ii) at each time step, the solution is -close to a steady solution.In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 – independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady – or quasi-steady – solution, which justifies the fact that the relevant time scale is ^-2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument. Mathematics Subject Classification (2000) Primary 80A25, Secondary 35K57, 47G20     

Ignition during hydrogen release from high pressure into the atmosphere

The first investigations concerned with a problem of hydrogen jet ignition, during outflow from a high-pressure vessel were carried out nearly 40 years ago by Wolanski and Wojcicki. The research resulted from a dramatic accident in the Chorzow Chemical Plant Azoty, where the explosion of a synthesis gas made up of a mixture composed of three moles of hydrogen per mole of nitrogen, at 300°C and 30 MPa killed four people. Initial investigation had excluded potential external ignition sources and the main aim of the research was to determine the cause of ignition. Hydrogen is currently considered as a potential fuel for various vehicles such as cars, trucks, buses, etc. Crucial safety issues are of potential concern, associated with the storage of hydrogen at a very high pressure. Indeed, the evidence obtained nearly 40 years ago shows that sudden rupture of a high-pressure hydrogen storage tank or other component can result in ignition and potentially explosion. The aim of the present research is identification of the conditions under which hydrogen ignition occurs as a result of compression and heating of the air by the shock wave generated by discharge of high-pressure hydrogen. Experiments have been conducted using a facility constructed in the Combustion Laboratory of the Institute of Heat Engineering, Warsaw University of Technology. Tests under various configurations have been performed to determine critical conditions for occurrence of high-pressure hydrogen ignition. The results show that a critical pressure exists, leading to ignition, which depends mainly on the geometric configuration of the outflow system, such as tube diameter, and on the presence of obstacles.