We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

Hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneousporous media. It critically depends on dimensionality. The asymptotic macrodispersion,transverse to the mean velocity direction, vanishes only in 2D and not in 3D. Using theclassical Gaussian correlated permeability fields with a lognormal distribution ofvariance σY2, thelongitudinal and transverse dispersivities are determined numerically as a function ofheterogeneity and dimensionality. We show that the transverse macrodispersion steeplyincreases with σY2 underlyingthe essential role of flow lines braiding, a mechanism specific to 3D systems. Thetransverse macrodispersion remains however at least two orders of magnitude smaller thanthe longitudinal macrodispersion, which increases even more steeply withσY2. At moderateto high levels of heterogeneity, the transverse macrodispersion also converges much fasterto its asymptotic regime than do the longitudinal macrodispersion. Braiding cannot be thustaken as the sole mechanism responsible for the high longitudinal macrodispersions. Itcould be either supplemented or superseded by stronger velocity correlations in 3D than in2D. This assumption is supported by the much larger longitudinal macrodispersions obtainedin 3D than in 2D, up to a factor of 7 for σY2 = 7.56.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

Plant growth depends essentially on nutrients coming from the roots and metabolitesproduced by the plant. Appearance of new branches is determined by concentrations ofcertain plant hormones. The most important of them are Auxin and Cytokinin. Auxin isproduced in the growing, Cytokinin in either roots or in growing parts. Many dynamicalmodels of this phenomena have been studied in [1]. In [5], the authors deal with onebranch model. In this work, we focus our interest on a multiple branch model. We deal withthe transport equation in domains of different sizes. A variational reduction type method[3] based on asymptotic partial decomposition introduced in [2] (see also [4]) is used. Inthis work we consider the transport equation in decomposed domain with a general righthand side

We study the mathematical properties of a general model of cell division structuredwith several internal variables. We begin with a simpler and specific model with two variables, wesolve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-timeconvergence. The main difficulty comes from natural degeneracy of birth terms that we overcomewith a regularization technique. We then extend the results to the case with several parameters andrecall the link between this simplified model and the one presented in [6]; an application to thenon-linear problem is also given, leading to robust subpolynomial growth of the total population.

We present a domain decomposition theory on an interface problemfor the linear transport equation between a diffusive and a non-diffusive region.To leading order, i.e. up to an error of the order of the mean free path in thediffusive region, the solution in the non-diffusive region is independent of thedensity in the diffusive region. However, the diffusive and the non-diffusive regionsare coupled at the interface at the next order of approximation. In particular, ouralgorithm avoids iterating the diffusion and transport solutions as is done in mostother methods — see for example Bal–Maday (2002). Our analysis is based instead on an accurate description of the boundarylayer at the interface matching the phase-space density of particles leaving thenon-diffusive region to the bulk density that solves the diffusion equation.

Sungo Bay (China) has a mean depth of 10 m, a total area of 140 km2 and is occupied by several types of aquaculture, whilst opening to the ocean. The production of scallops (Chlamys farreri) cultured on long lines is estimated to exceed 50 000 tonnes (total weight) per year. Selection of sites for scallop growth and determination of suitable rearing densities have become important issues. In this study, we focused on the local scale (e.g. 1000 m) where rearing density, food concentration and hydrodynamics interact. We have developed a depletion model coupling a detailed model of C. farreri feeding and growth and a one-dimensional horizontal transport equation. The model was applied to assess the effect of some environmental parameters (e.g. food availability, temperature, hydrodynamism) and spatial variability on growth, and to assess the effect of density according to a wide range of hydrodynamical and environmental conditions. In the simulations, food concentrations always enabled a substantial weight increase with a final weight above 1.5 g dry weight. Compared to a reference situation without depletion, a density of 50 ind m–3 decreased growth between 0% and 100%, depending on current velocity when maximum current velocity was below 20 cm s–1. The mean ratio between food available inside and outside the cultivated area (depletion factor) varied with the percentage of variation in scallop growth that was due to density. Our model suggests that scallop growth was correlated with maximum current velocity for a given density and current velocity below 20 cm s–1. The model was integrated within a Geographical Information System (GIS) to assist in making decisions related to appropriate scallop densities suitable for aquaculture at different locations throughout the bay. Concepts (depletion), methods (coupling hydrodynamics and growth models), and the underlying framework (GIS) are all generic, and can be applied to different sites and ecosystems where local interactions must be taken into account.