Optimal and better transport plans

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=∞} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a “better” notion of optimality called robust optimality.     

Evaluating the transport in small-world and scale-free networks

We present a study of some properties of transport in small-world and scale-free networks. Particularly, we compare two types of transport: subject to friction (electrical case) and in the absence of friction (maximum flow). We found that in clustered networks based on the Watts–Strogatz (WS) model, for both transport types the small-world configurations exhibit the best trade-off between local and global levels. For non-clustered WS networks the local transport is independent of the rewiring parameter, while the transport improves globally. Moreover, we analyzed both transport types in scale-free networks considering tendencies in the assortative or disassortative mixing of nodes. We construct the distribution of the conductance G and flow F to evaluate the effects of the assortative (disassortative) mixing, finding that for scale-free networks, as we introduce different levels of the degree–degree correlations, the power-law decay in the conductances is altered, while for the flow, the power-law tail remains unchanged. In addition, we analyze the effect on the conductance and the flow of the minimum degree and the shortest path between the source and destination nodes, finding notable differences between these two types of transport.     

Singular one-dimensional transport equations on Lp spaces

We prove the well-posedness of the Cauchy problem governed by a linear mono-energetic singular transport equation (i.e., transport equation with unbounded collision frequency and unbounded collision operator) with specular reflecting and periodic boundary conditions on L_p spaces. The large time behaviour of its solution is also considered. We discuss the compactness properties of the second-order remainder term of the Dyson-Phillips expansion for a large class of singular collision operators. This allows us to evaluate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived.     

CFD of turbulent transport of particles behind a backward-facing step using a new model—k-ε-Sp

The k-ε-S_p model, describing two-dimensional gas–solid two-phase turbulent flow, has been developed. In this model, the diffusion flux and slip velocity of solid particles are introduced to represent the particle motion in two-phase flow. Based on this model, the gas–solid two-phase turbulent flow behind a vertical backward-facing step is simulated numerically and the turbulent transport velocities of solid particles with high density behind the step are predicted. The numerical simulation is validated by comparing the results of the numerical calculation with two other two-phase turbulent flow models (k-ε-A_p, k-ε-k_p) by Laslandes and the experimental measurements. This model, not only has the same virtues of predicting the longitudinal transport of the solid particles as the present practical two-phase flow models, but also can predict the lateral transport of the solid particles correctly.     

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.     

1|1 parallel transport and connections

A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that every such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.     

Formulation of a Ritz-Galerkin type procedure for the approximate solution of the neutron transport equation

The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ(x, ω) over a domain G × S in the five-dimensional phase space $\text{E}{}_3\text{× S(¦ ω ¦ = 1)}$. In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace.     

Well-posedness for a transport equation with nonlocal velocity

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Córdoba, Córdoba and Fontelos [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in Hmax{3/2−γ,0}. While in the supercritical case, the problem is locally well-posed with initial data in H3/2−γ, and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389].     

Complex metabolic network of glycerol fermentation by Klebsiella pneumoniae and its system identification via biological robustness

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) can be characterized by an intricate metabolic network of interactions among biochemical fluxes, metabolic compounds, key enzymes and genetic regulation. Since there are some uncertain factors in the fermentation, especially the transport mechanisms of substances across cell membrane, the metabolic network contains multiple possible metabolic systems. In this paper, we establish a complex metabolic network and the corresponding nonlinear hybrid dynamical system aiming to determine the most possible metabolic system. The existence, uniqueness and continuity of solutions are discussed. We quantitatively describe biological robustness and present a system identification model on the basis of robustness performance. The identification problem is decomposed into two subproblems and a procedure is constructed to solve them. Numerical results show that it is most possible that both glycerol and 1,3-PD pass the cell membrane by active transport coupling with passive diffusion under substrate-sufficient conditions, whereas, under substrate-limited conditions, glycerol passes cell membrane by active transport coupling with passive diffusion and 1,3-PD by active transport.     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation

     

Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate

A coefficient inverse problem for the non-stationary single-speed transport equation for t∈(0,T) with the lateral boundary data and initial condition at t=0 is considered. Global uniqueness result is obtained by the method of Carleman estimates.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

Some applications of functional analysis to problems of neutron transport

Ritz method is used to obtain an approximate solution of the stationary neutron transport Boltzmann equation in its integral form in plane and spherical symmetry. Such a method is based on the maximum property of the quadratic form corresponding to a symmetric transformation in a finite dimensional subspace spanned by the firstn functions of a complete orthonormal set. In order to justify some numerical results, we also show that the transport operators are compact as acting both on the spaceC and on the spaceL ₂. Moreover, we investigate some properties of the solution and we prove that the neutron distribution is not increasing as a function of the spatial coordinate. Finally, a series of calculations have been performed for various values of the system-dimensions measured in mean-free-paths and the number of secondaries per conllision to maintain the system critical has been found.     

Deux remarques sur les flots généralisés d'équations différentielles ordinaires

This Note presents two remarks on the notion of generalized flow solution to ordinary differential equations, as introduced by DiPerna and the third author (R.J. Di Perna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (3) (1989) 511–547). On the one hand, we provide a self-contained proof of the uniqueness of such a flow. By this, we mean that our new proof does not exploit the interpretation of the generalized flow in terms of flow for the associated linear transport equation. On the other hand, this time using the associated linear transport equation, we slightly extend the result of uniqueness contained in the article cited, proving it holds without the group property of the flow (in the time variable). To cite this article: M. Hauray et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem

Numerical approximation of the coupled system of compressible miscible displacement problem in porous media is considered in this paper. A continuous in time discontinuous Galerkin scheme is developed. The symmetric interior penalty discontinuous Galerkin method is used to solve both the flow and transport equations. Upwind technique is used to treat the convection term in the transport equation. The hp-a priori error bounds are derived.     

A new numerical method for the solution of the Boltzmann equation in the semiconductor nonlinear electron transport problem

A new iterative method for solving the Boltzmann transport equation in the space-uniform case is introduced. The method is based on the use of a mesh in the momentum space to represent the distribution function. In intermediate points the function is found with the help of interpolation. The method is used to study the hot-electron transport in bulk n-InN, which is a promising material for optoelectronic applications.     

Duality for rectified cost functions

It is well-known that duality in the Monge–Kantorovich transport problem holds true provided that the cost function c : X × Y → [0, ∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification c _r of the cost c, so that the Monge-Kantorovich duality holds true replacing c by c _r . In particular, passing from c to c _r only changes the value of the primal Monge–Kantorovich problem. Finally, the rectified function c _r is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.     

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.     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Radiation transport in plane-parallel media with non-uniform surface illumination

Fourier and other integral transform techniques are used to reduce the problem of radiation transport in plane-parallel media with non-uniform surface illumination to a convenient computational form, and theF _N method is used to provide a basis for approximate solutions.