Continuity and Transport Equations : Theory, Numerics & Modeling

In this thesis, we study the continuity and the transport equations under the assumption of a non-regular vector field. We begin by reviewing the classical theory and the connection of these equations to the Lagrangian formulation of ordinary differential equations. After this introductory discussion, we develop the theory for the non-classical case, based on the seminal works of DiPerna-Lions and Ambrosio, and we propose an alternative, more geometric, method to prove the well-posedness of the equations. We then introduce an approximation scheme, based on Lagrangian quantitative estimates, for flows associated with non-regular velocity fields (i.e. regular Lagrangian flows), which allows us to design an explicit numerical method to approximate the solutions of the aforementioned equations. Such a method is based on a Lagrangian description of the equations and it has some advantages with respect to methods based on an Eulerian point of view, such as the lack of a CFL condition and the applicability to non-cartesian meshes. We then show that the numerical scheme we propose is suitable also for approximating the solution of an advection-diffusion equation, exploiting the link between the Lagrangian and Eulerian perspectives, which remains valid in this setting. Finally, we investigate the connection between stochastic reorientations of a velocity field and enhanced diffusion in a fluid. In particular, we are motivated by the real-world example of Stylonychia (a genus of ciliates), whose feeding activity creates flows which ultimately enhance the diffusivity of the fluid. We then develop a mathematical model to estimate the enhanced diffusion produced by a population of similar microorganisms, and we compare our results to experimental measurements.