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We describe a time-dependent functional involving the relative entropy and the P H1 seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/ blowup issues for the Landau equation with Coulomb potential. [ABSTRACT FROM AUTHOR]

Abstract This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada–Kawasaki–Teramoto (SKT) type (cf. [1]) satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation–fragmentation–diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng–Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [8] for the Vlasov–Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density. [ABSTRACT FROM AUTHOR]

Abstract: We consider in this paper the initial-boundary value problem for the 1D neutron transport equation with isotropic scattering, set in some bounded interval with inflow boundary conditions. The usual parabolic scaling yields the diffusive limit. A surrogate model, coupling transport and diffusion equations, is then introduced in order to accurately assess the value of specific quantities of interest. The control of the quality of the computation (with respect to such a quantity of interest) is performed by means of an approach for modeling error estimation combined with an adaptive strategy to control the surrogate model, if necessary. [Copyright &y& Elsevier]

Abstract: We study in this paper a few simple examples of hypocoercive systems in which the coercive part is degenerate. We prove that the (completely explicit) speed of convergence is at least of inverse power type (the power depending on the features of the considered system). [Copyright &y& Elsevier]

Abstract: In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reaction–diffusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative effects of diffusion and reaction are given. [Copyright &y& Elsevier]

Abstract: For the homogeneous Boltzmann equation with (cutoff or noncutoff) hard potentials, we prove estimates of propagation of norms with a weight (, large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means. [Copyright &y& Elsevier]

Abstract: We present the various levels of possible modeling for multiphase flows: coupling of fluid equations in different domains with a free boundary; coupling (in the same domain) of a fluid equation and a kinetic (Vlasov or Vlasov–Boltzmann) equation; coupling (in the same domain) of two (or more) fluid equations. We briefly present the mathematical results relative to the passage from one of these approaches to another approach, and we give some ideas of how to use those different models on a specific practical example. [Copyright &y& Elsevier]

The rigorous asymptotics from reaction-cross-diffusion systems for three species with known entropy to cross-diffusion systems for two variables is investigated. The equations are studied in a bounded domain with no-flux boundary conditions. The global existence of very weak (integrable) solutions and the rigorous fast-reaction limit are proved. The limiting system inherits the entropy structure with an entropy that is not the sum of the entropies of the components. Uniform estimates are derived from the entropy inequality and the duality method. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic (large time) behaviour of a selection–mutation–competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε of mutations. We show that depending on α > 0 , the limit ε → 0 with t = ε − α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large). [ABSTRACT FROM AUTHOR]