Investigations of the Dynamics of Models of Heat Transfer and Clustering

We investigate two models that are widely used in physics and engineering. Our main goal is to study conjectures made by physicists with full mathematical rigor.

The first model we investigate is the advection-diffusion equation. Here, we address the problem of optimizing heat transfer via an incompressible fluid in a bounded domain. We use techniques in probability theory to get bounds for the heat transfer rate. Asymptotically, we obtain matching lower and upper bounds (up to a logarithmic factor) over a class of velocity field of the fluid that satisfies an energy like constraint. This gives a rigorous proof for a result by Marcotte et. al. (SIAM Appl. Math ’18). We also give an upper bound for for the problem under an enstrophy-like constraint.

The second model is the coagulation-fragmentation equation, which models the evolution ofthe density particle sizes in a system where particles can split and merge. Depending on the coagulation and fragmentation kernels, solutions of the system will behave differently. Here, we address two problems. The first problem concerns the well-posedness of mass-conserving solutions when the coagulation kernel is multiplicative and the fragmentation kernel constant. This belongs to a so-called critical case, where existence of mass-conserving solutions depends on how large the system is initially. Here, we develop a new technique by studying properties of the viscosity solutions of a corresponding singular Hamilton-Jacobi equation to deduce information about the solutions to the coagulation fragmentation equation. Using this technique, we moved one step closer to resolving a long-standing conjecture in the field.

Still under the umbrella of the coagulation-fragmentation equation, we study the dynamics of the solution when the coagulation kernel is multiplicative and the fragmentation kernel additive and small. The problem we are concerned here resembles singular perturbation problems in PDEs. Letting the fragmentation kernel vanish, in the limit, one expects that the solutions tend to the so-called Flory solution of the pure multiplicative coagulation equation, where part of the total mass escapes to infinity. We study how the lost mass behaves. Our proposed idea is based on the study of a nonlinear backward parabolic equation, result from the Bernstein transform of the equation, and a detailed study of the tail behavior of the Flory solution of the pure coagulation equation with multiplicative kernel. With this idea, we made some progress towards resolving a prediction by Ben-Naim and Krapivsky (Phys. Rev. E 83, 061102, 2011).