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.     

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.     

Remarks on existence and uniqueness of Cournot–Nash equilibria in the non-potential case

This article is devoted to various methods (optimal transport, fixed-point, ordinary differential equations) to obtain existence and/or uniqueness of Cournot–Nash equilibria for games with a continuum of players with both attractive and repulsive effects. We mainly address separable situations but for which the game does not have a potential, contrary to the variational framework of Blanchet and Carlier (Optimal transport and Cournot–Nash equilibria, 2012). We also present several numerical simulations which illustrate the applicability of our approach to compute Cournot–Nash equilibria.     

Vorticity transport in a viscoelastic fluid in the presence of suspended particles through porous media

We consider the transport of vorticity in an Oldroydian viscoelastic fluid in the presence of suspended magnetic particles through porous media. We obtain the equations governing such a transport of vorticity from the equations of magnetic fluid flow. It follows from these equations that the transport of solid vorticity is coupled to the transport of fluid vorticity in a porous medium. Further, we find that because of a thermokinetic process, fluid vorticity can exist in the absence of solid vorticity in a porous medium, but when fluid vorticity is zero, then solid vorticity is necessarily zero. We also study a two-dimensional case.     

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.     

On the self-adjoint subspace of the one-velocity transport operator

We study the problem of describing the self-adjoint subspace of the transport operator in an unbounded domain. It is proved that this subspace is nontrivial under perturbations having a lattice of gaps of arbitrarily small length for the one-velocity operator with polynomial collision integral. We also consider the three-dimensional transport operator.     

一类具积分边界条件种群细胞迁移算子的本质谱

本文在L^1空间上,研究一类具积分边界条件种群细胞迁移方程,利用泛函分析中构造算子和比较算子方法及相关半群知识证明了迁移算子A_H产生的G_0半群V_H（t）的Dyson-Phillips展开式的n阶余项R_n（t）（n≥1）的弱紧性及V_H（t）和U_H（t）（streaming算子B_H产生）具有相同的本质谱及一致的本质谱型,得到了在区域Г中迁移算子A_H仅由有限个具有限代数重数的离散本征值组成及迁移方程解的渐近稳定性．     

Mass Transportation with LQ Cost Functions

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier’s Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

A Monge–Kantorovich mass transport problem for a discrete distance

This paper is concerned with a Monge-Kantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDE formulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how these problems, when rescaling the step distance, approximate the classical problem. In particular we obtain, taking limits in the rescaled nonlocal formulation, the PDE formulation given by Evans-Gangbo for the classical problem.     

Unconditionally stable schemes for convection-diffusion problems

Convection-diffusion problems are basic ones in continuum mechanics. The main features of these problems are connected with the fact that their operators may have an indefinite sign. In this paper we study the stability of difference schemes with weights for convection-diffusion problems where the convective transport operator may have various forms. We construct unconditionally stable schemes for nonstationary convection-diffusion equations based on the use of new variables. Similar schemes are also used for parabolic equations where the operator represents the sum of diffusion operators.     

SOME PROBLEMS IN RADIATION TRANSPORT FLUID MECHANICS AND QUANTUM FLUID MECHANICS

We introduce the radiation transport equations, the radiation fluid mechanics equations and the fluid mechanics equations with quantum effects. We obtain the unique global weak solution for the radiation transport fluid mechanics equations under certain initial and boundary values. In addition, we also obtain the periodic region problem of the compressible N-S equation with quantum effect has weak solutions under some conditions.     

A Planning Problem Combining Calculus of Variations and Optimal Transport

We consider some variants of the classical optimal transport where not only one optimizes over couplings between some variables x and y but also over some control variables governing the evolutions of these variables with time. Such a situation is motivated by an assignment problem of tasks with workers whose characteristics can evolve with time (and be controlled). We distinguish between the coupled and decoupled case. The coupled case is a standard optimal transport with the value of some optimal control problem as cost. The decoupled case is more involved since it is nonlinear in the transport plan.     

Probability density function of leaky integrate-and-fire model with Lévy noise and its numerical approximation

We investigate the numerical analysis of leaky integrate-and-fire model with Lévy noise. We consider a neuronal model in which probability density function of a neuron in some potential at any time is modeled by a transport equation. Lévy noise is included due to jumps by excitatory and inhibitory impulses. Due to these jumps the resulting equation is a transport equation containing two integral in right-hand side (jumps). We design, implement, and analyze numerical methods of finite volume type. Some numerical examples are also included.     

From the Schrödinger problem to the Monge–Kantorovich problem

The aim of this article is to show that the Monge–Kantorovich problem is the limit, when a fluctuation parameter tends down to zero, of a sequence of entropy minimization problems, the so-called Schrödinger problems. We prove the convergence of the entropic optimal values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the cluster points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Γ-convergence which we didn?t find in the literature; these Γ-convergence results which are interesting in their own right are also proved.     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

On weakly nonlinear waves in media exhibiting mixed nonlinearity

We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.