Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structureAnalyse de perturbation singulière pour la réduction de la chimie complexe des mélanges gazeux avec structure entropique

In this Note, we investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions, the equilibrium constant of which are compatible with thermodynamics, thus providing an entropic structure. We assume that a subset of the reactions is consituted of fast reactions and define a constant and linear projection onto the partial equilibrium manifold compatible with the entropy production. This reduction step is used for the study of a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. To cite this article: M. Massot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 93–98.     

Multicomponent flow modeling

We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations.We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves,also termed anchored waves.We further discuss related models which have a similar hyperbolic-parabolic structure,notably the SaintVenant system with a temperature equation as well as the equations governing chemical equilibrium flows.We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure.We finally discuss numerical algorithms specifically devoted to complex chemistry flows,in particular the evaluation of multicomponent transport properties,as well as the impact of multicomponent transport.     

Flammes planes avec transport multi-espèce et chimie complexe

We consider plane laminar flames with multicomponent transport and complex chemistry. The governing equations are derived from the kinetic theory of gases. An arbitrary number of reversible chemical reactions and temperature dependent species specific heats are considered in the model. The most general form for multicomponent transport fluxes given by the kinetic theory is also taken into account. Upon first considering a bounded domain and then letting the size of the domain to go to infinity, we obtain an existence theorem. A priori estimates fundamentally rely on the entropy which correlates the transport fluxes and the gradients.     

The stability of constant equilibrium states in relaxation models

We consider various first-order systems of PDEs with partial dissipation, as well as partial conservation. This class includes relaxation models, for instance the one designed by S. Jin and Z. Xin, as well as discrete velocity models for gases, as the Broadwell system. As we showed in a recent paper, the Jin-Xin model admits a convex compact positively invariant region, whenever the equilibrium system does. As a by-product, we obtained the existence of global weak solutions for the Cauchy problem with large data. For more general systems, the global existence of a uniformly bounded entropy solution will be a basic assumption in this work. We consider one-dimensional data which are either space periodic or square integrable. We prove that the (expected globally bounded) entropy solution relaxes to the equilibrium state; the latter is either zero or is determined by the mean value of the conserved components. We emphasize that we do not need any assumption about the nonlinearity of the underlying equilibrium system. We give two different proofs of the stabilization, which apply in different contexts. The first one uses compensated compactness and has a rather broad efficiency. For instance, it applies to several quasi-linear models. But the convergence result does not provide any decay rate in the periodic setting. The other one uses a dispersion estimate for the principal part of the model. It applies to periodic data and needs the strong assumption of semi-linearity, but yields an exponential decay in theL ²-norm. We expect that it could extend to multi-dimensional contexts.     

Une nouvelle méthode de relaxation pour les équations de Navier–Stokes compressibles

We consider the compressible Navier–Stokes equations for gas flows endowed with general pressure and temperature laws as long as they are compatible with the existence of an entropy and Gibbs relations. We extend the relaxation method introduced for the Euler equations by Coquel and Perthame. Keeping the same “sub-characteristic” conditions for the hyperbolic fluxes and using a consistent splitting of the diffusive fluxes based on a global temperature, we prove the stability of the relaxation system via the sign of the production of a suitable entropy. A first order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method. To cite this article: E. Bongiovanni et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the inequality of partial temperatures of a homogeneous gas mixture in the thermodynamic equilibrium state

A modified definition of the state of thermodynamic equilibrium of a homogeneous multicomponent gas mixture is given. It is based on the discrete character of collisions but not on the smoothed-by-multiple-collisions dynamic structure with the physical determination of the kinetic time dt. Application of rigorous mathematical methods which adequately describe the discrete character of collisions leads to a qualitatively new structure of the distribution functions and unexpected physical consequences. It is shown that owing to the inequality of head-on and head-tail collisions of particles, partial temperatures of a homogeneous multicomponent gas mixture in thermodynamic equilibrium cannot be equal to each other. It is found that relations of the partial temperatures to the mixture temperature are dependent on the molecular weights and concentrations of the gas components.     

Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.     

Multiscale measures of equilibrium on finite dynamic systems

This article presents a new method for the study of the evolution of dynamic systems based on the notion of quantity of information. The system is divided into elementary cells and the quantity of information is studied with respect to the cell size. We have introduced an analogy between quantity of information and entropy, and defined the intrinsic entropy as the entropy of the whole system independent of the size of the cells. It is shown that the intrinsic entropy follows a Gaussian probability density function (PDF) and thereafter, the time needed by the system to reach equilibrium is a random variable. For a finite system, statistical analyses show that this entropy converges to a state of equilibrium and an algorithmic method is proposed to quantify the time needed to reach equilibrium for a given confidence interval level. A Monte-Carlo simulation of diffusion of A* atoms in A is then provided to illustrate the proposed simulation. It follows that the time to reach equilibrium for a constant error probability, t_e, depends on the number, n, of elementary cells as: t_e∝n^2.22_±0.06. For an infinite system size (n infinite), the intrinsic entropy obtained by statistical modelling is a pertinent characteristic number of the system at the equilibrium.     

A note on the long time behavior for the drift-diffusion-Poisson system

In this note we analyze the long time behavior of a drift-diffusion-Poisson system with a symmetric definite positive diffusion matrix, subject to Dirichlet boundary conditions. This system models the transport of electrons in semiconductor or plasma devices. By using a quadratic relative entropy obtained by keeping the lowest order term of the logarithmic relative entropy, we prove the exponential convergence to the equilibrium. To cite this article: N. Ben Abdallah et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the generalization of conservation law theory to certain degenerate parabolic systems of equations describing processes of compressible two-phase multicomponent filtration

A degenerate parabolic system of equations of two-phase multicomponent filtration is considered. It is shown that this system can be treated as a system of conservation laws and the notions developed in the corresponding theory, such as hyperbolicity, shock waves, Hugoniot relations, stability conditions, Riemann problem, entropy, etc., can be applied to this system. The specific character of the use of such notions in the case of multicomponent filtration is demonstrated. An example of two-component mixture is used to describe the specific properties of solutions of the Riemann problem.     

On the stability of the equilibrium states for Hamiltonian dynamical systems arising in non–equilibrium thermodynamics

In this paper we consider a thermodynamic system with an internal state variable, and study the stability of its equilibrium states by exploiting the reduced entropy inequality. Remarkably, we derive a Hamiltonian dynamical system ruling the evolution of the system in a suitable thermodynamic phase space. The use of the Hamiltonian formalism allows us to prove the equivalence of the asymptotic stability at constant temperature, at constant entropy and at constant energy, thus extending some classical results by Coleman and Gurtin (J. Chem. Phys., 47, 597–613, 1967).     

The hydrodynamic equations for chemically equilibrium flows of a multielement plasma with exact transport coefficients

Explicit expressions for all of the effective transport coefficients are derived for thermochemically equilibrium flows using the exact mass and heat transfer equations, which are resolved with respect to the “forces” (the gradients of the hydrodynamic variables) via the flukes. It is shown that, in a mixture where the components have different diffusion properties, separation (diffusion) of the chemical elements occurs which leads to a state of affairs where the equilibrium concentrations, and together with them, the effective transport coefficients will be functions not only of the pressure and temperature but will also depend on the concentrations of the elements, determined when solving the problem (self-consistent concentrations of the elements). It is shown that the existence of an electric current and lack of quasineutrality (flow around electrically conducting walls—electrodes) does not change the structure of the expressions for the effective transport coefficients and does not add anything new. The approximate and incomplete treatment of thermochemically equilibrium flows of multicomponent gas mixtures and a plasma in previously published papers are especially noted. Numerical estimates of the effective transport coefficients are presented for an air plasma and the domains in the pressure-temperature plane with the required number of approximations in order to obtain results with an error of no worse than 5% are indicated.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. (Received 11 October 2000; in revised form 13 March 2001)     

Treatment Planning Optimization for Radiotherapy

We study the problem of treatment planning in radiotherapy. We use an energy-dependent Boltzmann-based particle transport model as a partial integro-differential constraint for a convex optimization problem. We solve the resulting optimal control problem using a minimum entropy moment closure. Numerical results for problems on patient data are presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear asymptotic stability of the equilibrium state for the MEP model of charge transport in semiconductors

Nonlinear asymptotic Lyapunov stability of the equilibrium state for the balance equations of charge transport in semiconductors based on the maximum entropy principle [A.M. Anile, V. Romano, Non parabolic band transport in semiconductors: closure of the moment equations, Contin. Mech. Thermodyn. 11 (1999) 307–325; V. Romano, Non parabolic band transport in semiconductors: closure of the production terms in the moment equations, Contin. Mech. Thermodyn. 12 (2000) 31–51] is proven for a typical 1-D problem.     

Linear asymptotic stability of the equilibrium state for the 2-D MEP hydrodynamical model of charge transport in semiconductors

Linear asymptotic Lyapunov stability of the equilibrium state for the balance equations of charge transport in semiconductors based on the maximum entropy principle [A.M. Anile, V. Romano, Non parabolic band transport in semiconductors: closure of the moment equations, Contin. Mech. Thermodyn. 11 (1999) 307–325; V. Romano, Non parabolic band transport in semiconductors: closure of the production terms in the moment equations, Contin. Mech. Thermodyn. 12 (2000) 31–51] is proven for a typical two-dimensional problem.     

On the entropy method and exponential convergence to equilibrium for a recombination–drift–diffusion system with self-consistent potential

We consider a Shockley–Read–Hall recombination–drift–diffusion model coupled to Poisson’s equation and subject to boundary conditions, which imply conservation of the total charge. As main result, we derive an explicit functional inequality between relative entropy and entropy production rate, which implies exponential convergence to equilibrium with explicit constant and rate. We report that the key entropy–entropy production inequality ought rather not to be formulated on the states space of the parabolic–elliptic system, but on the reduced states space of the associated nonlocal drift–diffusion problem, where the Poisson equation is replaced by the corresponding Green function.     

Hyperbolic conservation laws with stiff relaxation terms and entropy

We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.