Asymptotic derivation of diffusion models for mixtures

Project M3007N Asymptotic derivation of diffusion models for mixtures Abstract The Newtons second law of motion, the Boltzmann equation and the Euler and Navier-Stokes models describe at different physical scales the same underlying phenomenon, the interaction be- tween particles. To understand the mathematical coherence between these three regimes (micro- scopic, mesoscopic and macroscopic) is part of the well-known Hilberts sixth problem, which gave birth to a wide variety of fundamental results over the course of the last century and remains nowadays a fruitful and active field of research. Starting from a mesoscopic formulation, we intend to build a hierarchy of multi-species fluid equations for monatomic non-reactive mixtures, in order to establish the regimes of their mathe- matical validity and to compute the relevant transport coefficients. We want to study the strong convergence of solutions of the multi-species Boltzmann equation towards solutions of some suitable incompressible Navier-Stokes equations for mixtures (1) and then analyse the successive orders in the hydrodynamic approximation to show that the Maxwell-Stefan equations can be seen as a cor- rection to the multicomponent Ficks model (2). Finally, we shall tackle the problem of designing a suitable asymptotic-preserving scheme to capture the correct diffusion limit (3). Objectives (1) and (2) will be treated by first adapting the approach of Ellis and Pinsky to study the semigroup generated by the linearized multi-species Boltzmann operator and then by linking the Maxwellian states of equilibrium corresponding to Maxwell-Stefan, Navier-Stokes and Fick regimes. The approach will also involve the use of recent analytic techniques like hypocoercive and entropy methods to study the Cauchy problems for the kinetic and the macroscopic equations. To achieve goal (3) our idea is to combine the use of the micro-macro decomposition of Lemou and Mieussens and the penalization-based method of Filbet and Jin to find a suitable mesoscopic-macroscopic reformulation and to handle the stiffness of the collision operator. Our goals would extend fundamental results known to hold for mono-species hydrodynamic limits (allowing to recover a first rigorous derivation of the incompressible Navier-Stokes equations for mixtures) and would solve one of the oldest problems in multicomponent fluid mechanics, the controversy between the use of the Maxwell-Stefan or the Fick approach for modelling diffusion in mixtures. The main researchers involved at the University of Graz would be Klemens Fellner (reaction- diffusion systems, kinetic equations, cross-diffusion models, entropy methods) and Bao Quoc Tang (reaction-diffusion systems, entropy methods).

Mathematical models have historically proven useful to provide a satisfactory description of several real-life phenomena, that are connected with the investigations of other scientific disciplines. The work of an applied mathematician consists in proving that these models are sound, in the sense that they can be deduced from basic physical principles or from natural conservation laws, and that they satisfy a series of fundamental properties (for example the fact that a certain system of equations can indeed be solved and does not lead to unphysical phenomena). The project "Asymptotic derivation of diffusion models for mixtures" belongs to the research field of kinetic theory of gases, proposed by Ludwig Boltzmann, and concerns the study of mathematical models that describe the dynamics of interacting multi-species particle systems. The typical application example is fluid mixtures, in which molecules of different species influence each other through microscopic collisions or chemical reactions, and determine the macroscopically observable behaviours. Depending on the scale of observation of the phenomenon, different models can be proposed, and it is important to deduce mathematically rigorous equations for the evolution of the observable physical quantities (like the density, the temperature or the pressure of the gas) from the microscopic level. In this context, the main outcome of the project was the ability to establish new links between the solutions of the Boltzmann equation for a gas mixture with those of the Maxwell-Stefan and reaction-diffusion systems, so as to determine the physical regime of transition between the two models. The question is of significance from a theoretical and an applicative point of view. Indeed, on the one side these analyses can define a unified mathematical framework where all these different models are linked together, proving that they are in fact coherent with one another. On the other, we deal with models that can be generalized to describe interactions between agents, much more complex than the collisions. One can think, for example, of a population of animals belonging to different species that interact in a natural habitat by competing for food or through predator-prey relationships, or of the problem of modelling the spread of an epidemic inside a human population, where the different individuals (that may or may not be infected with the virus) interact via social contacts. A study of these two extensions is ongoing, as a natural continuation of the Researcher's work. Therefore, the outcomes obtained by the project have immediate implications in the specific area of kinetic theory of gases, but they may also provide in the future a bridge between different mathematical and non-mathematical scientific communities, fostering the way to the development of possible new interesting interdisciplinary collaborations, including applications in physics, chemistry, biology and medicine.