anisotropic **convection**-**diffusion** **equations** set in a bounded bidimensional domain and endowed with either Dirichlet and / or no-flux boundary conditions. We show that solutions to the two-point flux approximation (TPFA) and discrete duality finite volume (DDFV) schemes under consideration converge exponentially fast toward their steady state. The analysis relies on discrete entropy estimates and discrete functional inequalities. As a biproduct of our analysis, we establish new discrete Poincaré-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our theoretical results are illustrated by numerical simulations.

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Abstract. We are interested in the large-time behavior of solutions to finite volume dis- cretizations of **convection**-**diffusion** **equations** or systems endowed with non-homogeneous Dirich- let and Neumann type boundary conditions. Our results concern various linear and nonlinear models such as Fokker-Planck **equations**, porous media **equations** or drift-**diffusion** systems for semiconductors. For all of these models, some relative entropy principle is satisfied and im- plies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserve this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centered convections or Scharfetter-Gummel discretizations. We illustrate our theoretical results on several numerical test cases.

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Abstract We propose a finite volume scheme for **convection**-**diffusion** **equations** with nonlinear **diffusion**. Such **equations** arise in numerous physical contexts. We will particularly focus on the drift-**diffusion** system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter-Gummel scheme for nonlinear **diffusion**. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

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Abstract
We study the existence of global weak solutions in a three-dimensional time-dependent bounded domain for the incompressible Vlasov-Navier- Stokes system which is coupled with two **convection**-**diffusion** **equations** describing the air temperature and its water vapor mass fraction. This newly introduced model describes respiratory aerosols in the human air- ways when one takes into account the hygroscopic effects, also inducing the presence of extra variables in the aerosol distribution function, tem- perature and size. The mathematical description of these phenomena leads us to make the assumption that the initial distribution of particles does not contain arbitrarily small particles. The proof is based on a reg- ularization and approximation strategy that we solve by deriving several energy estimates, including ones with temperature and size.

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[34, 35], for drift-**diffusion** systems [19, 7]. These last works show that the Scharfetter-Gummel numerical fluxes, first introduced in [42] for the approximation of **convection**-**diffusion** fluxes and widely used later for the simulation of semiconductor devices, preserve the thermal equilibrium. Their use in the numerical approximation of drift-**diffusion** systems ensure the exponential decay towards equilibrium of the numerical scheme. Unfortunately, such numerical fluxes can only be applied in two-points flux approximation finite volume schemes and therefore on restricted meshes. Moreover, they do not extend to anisotropic **convection**-**diffusion** **equations**.

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MSC (2010): 65M08, 35B40. 10
1 Motivation
We are interested in the numerical discretization of linear anisotropic **convection**- **diffusion** **equations** on almost general meshes. Let Ω be a polygonal connected open bounded subset of R 2 and let T > 0. The boundary Γ = ∂ Ω is divided into two parts Γ = Γ D ∪ Γ N with m(Γ D ) > 0. The problem writes:

Nous ´etudions le comportement asymptotique, lorsque t → ∞, des solutions de masse nulle du probl`eme de Cauchy pour l’´equation de **convection**-diﬀusion (1)-(2). Il est clair que cette ´equation met en jeu une comp´etition entre le terme de diﬀusion u xx et le terme de **convection** non lin´eaire (|u| q ) x , et
le but de notre ´etude est de d´eterminer lequel de ces deux termes devient pr´epond´erant lorsque t → ∞. Lorsque la masse totale M = R

Acknowledgements The authors are supported by the Inria teams RAPSODI and COFFEE, the LabEx CEMPI (ANR-11-LABX-0007-01) and the GEOPOR project (ANR-13-JS01-0007-01).
References
1. Andreianov, B., Bendahmane, M., Karlsen, K.H.: Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic **equations**. J. Hyperbolic Differ. Equ. 7(1), 1–67 (2010)

that this problem cannot be solved in the case of general bounded domain. However, it is getting feasible in some special cases when the periodic structure agrees with the geometry of the boundary of Ω. In Paper G we consider two such cases. Namely, we
study a **convection**-**diffusion** models in a thin rod and in a layer in R d . In the case of a thin rod we impose homogeneous Neumann boundary conditions on the lateral boundary of the rod and homogeneous Dirichlet boundary conditions on its bases. As was noticed above, the solution vanishes for time t ≫ ε. We determine the rate of vanishing of the solution and describe the evolution of its profile. If the effective axial drift is not zero (we study only this case), the rescaled solution concentrates in the vicinity of one of the rod ends, and the choice of the end depends on the sign of the effective **convection**. In order to characterize the rate of decay we introduce a 1-parameter family of auxiliary cell spectral problems (see [7], [10]). The asymptotic behaviour of the solution is then governed by the first eigenpair of the said spectral problem. Among the technical tools used in the paper, are factorization principle (see [23], [54], [56], [10], [11]), dimension reduction arguments and qualitative results required for constructing boundary layer correctors.

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The formula for the two-point flux at a given cell interface only involves the degrees- of-freedom of the two cells adjacent to that interface, thus offering the advantage of a very compact computational stencil. Nonetheless, a consistent formulation relies on the notion of admissible meshes, which requires some rather restrictive orthogonality constraint [21]. Combined with a first order **convection** flux, this approach is applied to the numerical discretization of non-coercive **convection**-**diffusion** **equations** in [19]. Several variants are also shown effective in the computation of anisotropic **diffusion** problems [20], in modeling groundwater flows in partially saturated porous media [25] and have been recently extended to general meshes by using some cell-based reconstruction of the solution gradient [22–24].

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Chapitre 5
Conclusion et perspectives
Dans ce mémoire, nous nous sommes intéressés à l’estimation d’erreur a pos- teriori pour l’équation de **convection**–**diffusion**–réaction linéaire et instationnaire discrétisée en espace par la méthode des volumes finis centrés par mailles et le schéma d’Euler implicite en temps. Ces estimations, calculées dans la norme d’éner- gie, sont basées sur trois idées. La solution du schéma volumes finis étant constante par morceaux en espace, la première idée consiste à la post-traiter localement. Ce post-traitement donne une solution polynômiale par morceaux en espace. L’erreur sera alors calculée par rapport à cette solution. La méthode des volumes finis étant localement conservative, la deuxième idée consiste à reconstruire des flux diffusif et convectif H(div, Ω)-conformes en utilisant les flux provenant du schéma volumes finis. Finalement, comme la solution post-traitée n’est pas conforme en espace, la troisième idée consiste à reconstruire un potentiel H 0 1 (Ω)-conforme. Le but de nos estimations d’erreur a posteriori est double : permettre le contrôle global de l’erreur par le biais de bornes calculables et fournir des indicateurs permettant le raffine- ment adaptatif en espace et en temps. Nous avons proposé un algorithme adaptatif basé sur ces estimations et présenté des essais numériques qui montrent l’efficacité de la stratégie adaptative. L’estimation en norme d’énergie ne fournissant pas une estimation d’erreur a posteriori robuste en régime de **convection** dominante, nous avons dérivé une nouvelle estimation d’erreur mesurée dans la norme d’énergie aug- mentée d’une norme duale de la dérivée en temps et de la partie anti-symétrique de l’opérateur différentiel. La nouvelle estimation est robuste en régime de **convection** dominante.

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We consider a family of Gagliardo-Nirenberg-Sobolev interpolation in- equalities which interpolate between Sobolev’s inequality and the logarith- mic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) mea- sures a distance to the manifold of the optimal functions. We give an ex- plicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution **equations** and improved entropy - entropy production estimates along the associated flow. Optimizing a relative entropy func- tional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution **equations**, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.

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5.1 Introduction
5.1.1 Setting and main results
Motivations Reaction-**diffusion** **equations** have drawn a lot of attention from the mathematical community over the last decades, but most usually in spatially homogeneous setting, while the lit- erature devoted to spatially heterogeneous domains only started developing recently. This growing interest led to many interesting questions regarding the possible effects of spatial heterogeneity on, for instance, the dynamics of the equation, or on optimization and control problems: how do these heterogeneities impact the dynamics or the criteria under consideration? Can the results obtained in the homogeneous case be obtained in the heterogeneous one, which is more relevant for applications? In this article, we study some of these questions and the influence of spatial heterogeneity from the angle of control theory. Some of our proofs and results are, however, of independent interest for reaction-**diffusion** **equations**.

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Abstract Within an averaging approach, the governing equations and effective boundary conditions describing both the average and pulsation motion of a near-critical fluid subjected to hi[r]

Abstract Within an averaging approach, the governing equations and effective boundary conditions describing both the average and pulsation motion of a near-critical fluid subjected to hi[r]

1.1 Mathematical model
Throughout the entire paper we will assume that all species diffuse with the same speed, i.e. for all i ∈ {1, · · · , q}, L i = L, where L is general **diffusion** generator. (While this
is a restrictive case - as generally this hypothesis is not realistic - assuming identical **diffusion** will allow us to obtain some optimal bounds.)

The second cell is hanged by the cold plate, whose temperature is ﬁxed by a regulating water bath. Then, nothing except air is in contact with the hot plate. We could not hang the ﬁrst cell the same way. Its hot plate ﬁts into plastic supports. The ﬁrst cell is further sur- rounded by a copper screen. This screen is temperature regulated at the average temperature between top and bottom plates, which is maintained constant on a whole set of measurements. In addition, both cells are wrapped in isolating sheets to limit air **convection**.

For time heterogeneous and space homogeneous bistable **equations**, the existence and qualitative properties of standard traveling, pulsating or transition fronts have been investigated by Alikakos, Bates and Chen [2], Fang and Zhao [18], and Shen [44, 45, 46, 47]. But, for space heterogeneous bistable **equations**, there is not so much work on transition fronts. In periodic media, the existence of pulsating fronts for (1.1), under the assumption (1.2), have just been discussed in our previous paper [16]. More precisely, under various additional assumptions on the reaction terms f (x, u) and by using different types of arguments, we proved several existence results of pulsating fronts with nonzero or zero speeds when the spatial period L is small or large (we will come back to the precise assumptions in the comments following Theorem 1.5 below). We also established some properties of the set of periods for which there exist pulsating fronts with nonzero speeds. However, in a given periodic medium, finding a general necessary and sufficient condition for the existence of pulsating fronts with nonzero speeds is still unclear in general. We point out that the existence result is known to hold for all L > 0 in some particular cases where f = f (u) does not depend on x and a L is close to

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E-mail: karch@math.uni.wroc.pl http://www.math.uni.wroc.pl/∼karch
Abstract. This note is devoted to the study of the long time behaviour of the solutions to the heat and the porous medium **equations** in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csisz´ ar-Kullback type estimates.

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