Browsing by Author "Vuoksenmaa, Aleksis Ilari"

Coagulation equations are evolution equations that model the time-evolution of the size-distribution of particles in systems where colliding particles stick together, or coalesce, to form one larger particle. These equations arise in many areas of science, most prominently in aerosol physics and the study of polymers. In the former case, the colliding particles are small aerosol particles that form ever larger aerosol particles, and in the latter case, the particles are polymers of various sizes. As the system evolves, the density of particles of a specified size changes. The rate of change is specified by two competing factors. On one hand there is a positive contribution coming from smaller particles coalescing to form particles of this specific size. On the other hand, particles of this size can coalesce with other particles to form larger particles, which contributes negatively to the density of particles of this size. Furthermore, if there is no addition of new particles into the system, then the total mass of the particles should remain constant. From these considerations, it follows that the time-evolution of the coagulation equation is specified for every particle size by a difference of two terms which preserve the total mass of the system. The physical properties of the system affect the time evolution via a coagulation kernel, which determines the rate at which particles of different sizes coalesce. A variation of coagulation equations is achieved when we add an injection term to the evolution equation to account for new particles injected into the system. This results in a new evolution equation, a coagulation equation with injection, where the total mass of the system is no longer preserved, as new particles are added into the system at each point in time. Coagulation equations with injection may have non-trivial solutions that are independent of time. The existence of non-trivial stationary solutions has ramifications in aerosol physics, since these might map to observations that the particle size distribution in the air stays approximately constant. In this thesis, it will be demonstrated, following Ferreira et al. (2019), that for any good enough injection term and for suitably picked, compactly supported coagulation kernels, there exists a stationary solution to a regularized version of the coagulation equation. This theorem, which relies heavily on functional analytic tools, is a central step in the proof that certain asymptotically well-behaved kernels have stationary solutions for any prescribed compactly supported injection term.