Convergence to Global Equilibrium for Spatially Inhomogeneous Kinetic Models of Non-Micro-Reversible Processes

We study the long-time asymptotics of linear kinetic models with periodic boundary conditions or in a rectangular box with specular reflection boundary conditions. An entropy dissipation approach is used to prove decay to the global equilibrium under some additional assumptions on the equilibrium distribution of the mass preserving scattering operator. We prove convergence at an algebraic rate depending on the smoothness of the solution. This result is compared to the optimal result derived by spectral methods in a simple one dimensional example.     

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.     

Strong solutions for a 1D viscous bilayer shallow water model

In this paper, we consider a viscous bilayer shallow water model in one space dimension that represents two superposed immiscible fluids. For this model, we prove the existence of strong solutions in a periodic domain. The initial heights are required to be bounded above and below away from zero and we get the same bounds for every time. Our analysis is based on the construction of approximate systems which satisfy the BD entropy and on the method developed by A. Mellet and A. Vasseur to obtain the existence of global strong solutions for the one dimensional Navier–Stokes equations.     

Systems of partial differential equations of mixed hyperbolic-parabolic type

We prove the global existence and uniqueness of solutions of certain mixed hyperbolic-parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier-Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new.     

Thermodynamic formalism for countable to one Markov systems

For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.

     

Compressible fluids and active potentials

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.     

A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

Global Periodic Attractor for Strongly Damped and Driven Wave Equations

In this paper we consider the strongly damped and driven nonlinear wave equations under homogeneous Dirichlet boundary conditions. By introducing a new norm which is equivalent to the usual norm, we obtain the existence of a global periodic attractor attracting any bounded set exponentially in the phase space, which implies that the system behaves exactly as a one dimensional system.     

Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division.Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.     

Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses

In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. We show the existence of a bounded positive invariant and attracting set. The possibility of existence and uniqueness of positive equilibrium are considered. The asymptotic behavior of the positive equilibrium and the existence of Hopf-bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. The existence and non-existence of periodic solutions are established under suitable conditions. The permanence conditions are also established. We obtained sufficient conditions to ensure the global stability of the unique positive equilibrium, by using suitable Lyapunov functions, LaSalle invariance principle and Dulac’s criterion. We obtained also sufficient conditions for the global stability of the prey-extinction equilibrium when the unique positive equilibrium is not feasible. Finally, numerical simulations are presented to illustrate the analytical results.     

Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry

In this paper, we investigate the system of equations modelling multicomponent reactive flows with detailed transport and complex chemistry in the limit of partial equilibrium. The reduced system is obtained using a projection step compatible with the chemical entropy production. The reduced multicomponent transport and convection fluxes are shown to be compatible with the mathematical entropy thus providing a symmetric form as well as normal forms for the reduced system. This yields global existence and asymptotic stability around constant equilibrium states for the Cauchy problem on the partial equilibrium manifold in all space dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

Entropies and weak solutions of the compressible isentropic Euler equations

In this Note, we study the system of isentropic Euler equations for compressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the density approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these results, we deduce the existence, compactness, and asymptotic decay of measurable and bounded entropy solutions.     

On Nash–Cournot oligopolistic market equilibrium models with concave cost functions

We consider Nash–Cournot oligopolistic market equilibrium models with concave cost functions. Concavity implies, in general, that a local equilibrium point is not necessarily a global one. We give conditions for existence of global equilibrium points. We then propose an algorithm for finding a global equilibrium point or for detecting that the problem is unsolvable. Numerical experiments on some randomly generated data show efficiency of the proposed algorithm.     

Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients

We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error criteria and then by applying a localization technique for one sample path on a bounded set.     

有界域上具有部分耗散和磁扩散的二维磁流体方程的全局适定性

探究了具有部分耗散和磁扩散的二维不可压缩磁流体(MHD)方程的初边值问题.在有界区域上,当系统的各个方向上的耗散系数和磁扩散系数都非负时,我们得到了该模型的强解是整体存在且唯一的.此外,对周期域而言,其解仍是全局适定的.     

Solution globale des equations de Maxwell-Dirac-Klein-Gordon

We prove the global existence on Minkowski space time of a solution of the Cauchy problem for the non linear system of coupled Maxwell, Dirac and Klein-Gordon equations, for small data with appropriate decay at space-like infinity. The method uses the conformal mapping of Minkowski space time onto a bounded open set of the Einstein cylinder.     

Global existence and open loop exponential stabilization of weak solutions for nonlinear Schrödinger equations with localized external Neumann manipulation

In this paper, we consider the nonlinear Schrödinger equations (NLS) with (focusing/defocusing) interior source and (possibly nonlinear) damping on a bounded regular domain in the Euclidean space. Moreover, it is assumed that the solutions are subject to external Neumann boundary manipulation on one portion of the boundary. Our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities. In the case of a linear damping, we also prove that solutions decay exponentially under the assumption that the Neumann type control at the boundary decays in a similar manner.     

Coupled system of Korteweg-de Vries equation-type in domains with oscillatory moving boundaries

We consider the initial-boundary value problem in a bounded domain with oscillatory moving boundaries and nonhomogeneous boundary conditions for the coupled system of equations of KdV type modelling strong interactions between internal solitary waves. We give a result of global existence and uniqueness for strong solutions for the coupled system of equations of Korteweg – de Vries type as well as the exponential decay of small solutions in asymptotically cylindrical domains. We present a numerical examples based in semi-implicit finite differences showing the numerical effect of the oscillatory moving boundaries for this kind of systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conditions at infinity for boltzmann's equation.

We consider here realistic conditions at infinity for solutions of the Boltzmann's equation, such as a pure Maxwellian equilibrium at infinity possibly with suitable boundary conditions on an exterior domain, different Maxwellian equilibria at +∞ and -;∞ in a tube-like situation and more generally conditions at infinity obtained from a fixed solution. In order to adapt the recent global existence and compactness results due to R.J. DiPerna and the author, we have to obtain some local a priori estimates on the mass, kinetic energy and entropy. And this is precisely what we achieve here by two different and new methods. The first one consists in using the relative entropy of solutions with respect to a fixed, possibly local, Maxwellian. This method allows to treat general collision kernels with angular cut-off and some of the conditions at infinity mentioned above. The second method is based upon a L¹ estimate and an extension of the entropy identity which uses a truncated H-functional. This method requires a “uniform integrability” condition on the collision kernel but allows to consider the most general conditions at infinity.     

Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.