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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]

We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds (with explicit rate). We show in this paper that this exponential convergence also holds when the reaction happens only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity. [ABSTRACT FROM AUTHOR]

The global existence of a chemotaxis model for cell aggregation phenomenon is obtained. The model system belongs to the class of logarithmic models and takes a Fokker–Planck type diffusion for the equation of cell density. We show that weak solutions exist globally in time in dimensions n ∈ { 1 , 2 , 3 } and for large initial data. The proof covers the parameter regimes that constant steady states are linearly stable. It also partially covers the other parameter regimes that constant steady states are unstable. We also find the sharp instability condition of constant steady states and provide numerical simulations which illustrate the formation of aggregation patterns. [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]

The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied using the socalled entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of the paper, complex balanced systems with boundary equilibria are considered. We focus on a specific case involving three chemical substances for which the boundary equilibrium is shown to be unstable in some sense, so that exponential convergence to the unique strictly positive equilibrium is recovered. [ABSTRACT FROM AUTHOR]

This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation D( f) by weighted relative Fisher information of f with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at-most linearly growing in-time rate. As an application of these estimates, we show the convergence of any ( H- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher L -moment. More precisely, if the initial data have some (large enough) polynomial L -moment, then we obtain an algebraic decay. If the initial data have a stretched exponential L -moment, then we recover a stretched exponential decay. [ABSTRACT FROM AUTHOR]

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose. [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]

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat, Collège de France Seminar, vol. II. Paris 1979–1980; vol. 60 of Res. Notes Math. Pitman, Boston, London (1982) 98–138]. [ABSTRACT FROM AUTHOR]

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level. [ABSTRACT FROM AUTHOR]

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another. [ABSTRACT FROM PUBLISHER]

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incompressible Navier-Stokes equation and of the Vlasov-Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations, and a compactness lemma specifically tailored for our study. [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]

The mechanism leading to gas damping in micro-electro-mechanical systems (MEMS) devices vibrating at high frequencies is investigated by using the linearized Boltzmann equation based on simplified kinetic models and diffuse reflection boundary conditions. Above a certain frequency of oscillation, the sound waves propagating through the gas are trapped in the gaps between the moving elements and the fixed boundaries of the microdevice. In particular, we found a scaling law, valid for all Knudsen numbers Kn (defined as the ratio between the gas mean free path and a characteristic length of the gas flow), that predicts a resonant response of the system. This response enables a minimization of the damping force exerted by the gas on the oscillating wall of the microdevice. [ABSTRACT FROM AUTHOR]

This paper is devoted to the establishment at the formal level of a scaling leading from gas-particles (also called Eulerian-Lagrangian) models for monodisperse thick sprays towards multiphase flows models (also called Eulerian-Eulerian) including the volume fraction as an unknown. This passage to the limit involves the study of an inelastic collision operator acting on internal energies as well as velocities. [ABSTRACT FROM AUTHOR]

We study a continuous coagulation-fragmentation model with constant kernels for reacting polymers (see [M. Aizenman and T. Bak, Comm. Math. Phys., 65 (1979), pp. 203-230]). The polymers are set to diffuse within a smooth bounded one-dimensional domain with no-flux boundary conditions. In particular, we consider size-dependent diffusion coefficients, which may degenerate for small and large cluster-sizes. We prove that the entropy-entropy dissipation method applies directly in this inhomogeneous setting. We first show the necessary basic a priori estimates in dimension one, and second we show faster-than-polynomial convergence toward global equilibria for diffusion coefficients which vanish not faster than linearly for large sizes. This extends the previous results of [J.A. Carrillo, L. Desvillettes, and K. Fellner, Comm. Math. Phys., 278 (2008), pp. 433-451], which assumes that the diffusion coefficients are bounded below. [ABSTRACT FROM AUTHOR]

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]

We propose a model based on the coupling of two Boltzmann-like equations for the study of the evolution of dust particles in a rarefied atmosphere, such as it can be found in the context of safety studies for the ITER project of nuclear fusion. When the typical size of a dust speck becomes too large, the numerical simulation of the system under study becomes too expensive, and one needs to introduce an asymptotic model in which the mass ratio between molecules and dust specks tends to 0. This model is constituted of a coupling (by a drag force term) between a Boltzmann equation and a Vlasov equation. A rigorous proof of the passage to the limit is given in the spatially homogeneous setting. It includes a new variant of Povzner’s inequality in which the vanishing mass ratio is taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions. [ABSTRACT FROM AUTHOR]

In this work, we consider the smoothness of the solutions to the full Landau equation. In particular, we prove that any classical solutions (such as the ones obtained by Guo in a “close to equilibrium” setting) become immediately smooth with respect to all variables. This shows that the Landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation. [ABSTRACT FROM AUTHOR]

We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6 π ν j where j is the current density of the particles, and of a friction term 6 π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/ N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure $\frac{1}{N}\sum_{1\le k\le N}\delta_{x_{k},v_{k}}$ , where x k is the center of the k-th ball and v k its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, []] who considered the case of periodically distributed x k s with v k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays. [ABSTRACT FROM AUTHOR]

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in L1((1+|v|s) dv) and Hk norms, s, k≥0 hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in L1 obtained by the so-called “entropy–entropy production” method in the context of dissipative systems with slowly growing a priori bounds (G. Toscani and C. Villani, J. Statist. Phys. 98 (2000), 1279–1309). [ABSTRACT FROM AUTHOR]

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]

Sprays are complex flows made of liquid droplets surrounded by a gas. They can be modeled by introducing a system coupling a kinetic equation (for the droplets) of Vlasov type and a (Euler-like) fluid equation for the gas. In this paper, we prove that, for the so-called thin sprays, this coupled model is well-posed, in the sense that existence and uniqueness of classical solutions holds for small time, provided the initial data are sufficiently smooth and their support have suitable properties. [ABSTRACT FROM AUTHOR]

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]

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space T(RN) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role. [ABSTRACT FROM AUTHOR]

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]

An enduring puzzle in evolutionary biology is to understand how individuals and populations adapt to fluctuating environments. Here we present an integro-differential model of adaptive dynamics in a phenotype-structured population whose fitness landscape evolves in time due to periodic environmental oscillations. The analytical tractability of our model allows for a systematic investigation of the relative contributions of heritable variations in gene expression, environmental changes and natural selection as drivers of phenotypic adaptation. We show that environmental fluctuations can induce the population to enter an unstable and fluctuation-driven epigenetic state. We demonstrate that this can trigger the emergence of oscillations in the size of the population, and we establish a full characterisation of such oscillations. Moreover, the results of our analyses provide a formal basis for the claim that higher rates of epimutations can bring about higher levels of intrapopulation heterogeneity, whilst intense selection pressures can deplete variation in the phenotypic pool of asexual populations. Finally, our work illustrates how the dynamics of the population size is led by a strong synergism between the rate of phenotypic variation and the frequency of environmental oscillations, and identifies possible ecological conditions that promote the maximisation of the population size in fluctuating environments. [ABSTRACT FROM AUTHOR]

Species endangered by rapid climate change may persist by tracking their optimal habitat; this depends on their dispersal characteristics. The Emperor penguin (EP) is an Antarctic seabird threatened by future sea ice change, currently under consideration for listing under the US Endangered Species Act. Indeed, a climate-dependent-demographic model without dispersion projects that many EP colonies will decline by more than 50% from their current size by 2100, resulting in a dramatic global population decline. Here we assess whether or not dispersion could act as an ecological rescue, i.e. reverse the anticipated global population decline projected by a model without dispersion. To do so, we integrate detailed dispersal processes in a metapopulation model—specifically, dispersal stages, dispersal distance, habitat structure, informed dispersal behaviors, and density-dependent dispersion rates. For EP, relative to a scenario without dispersion, dispersal can either offset or accelerate climate driven population declines; dispersal may increase the global population by up to 31% or decrease it by 65%, depending on the rate of emigration and distance individuals disperse. By developing simpler theoretical models, we demonstrate that the global population dynamic depends on the global landscape quality. In addition, the interaction among dispersal processes - dispersion rates, dispersal distance, and dispersal decisions - that influence landscape occupancy, impacts the global population dynamics. Our analyses bound the impact of between-colony emigration on global population size, and provide intuition as to the direction of population change depending on the EP dispersal characteristics. Our general model is flexible such that multiple dispersal scenarios could be implemented for a wide range of species to improve our understanding and predictions of species persistence under future global change. [ABSTRACT FROM AUTHOR]