ON THE WELL-POSEDNESS AND SELF-SIMILARITY OF CONTINUOUS NONLINEAR FRAGMENTATION MODELS

Abstract

This thesis addresses the fundamental question of the existence, uniqueness and nonexistence of mass-conserving weak solutions to the nonlinear fragmentation equation (NFE). It also investigates the existence of self similar solutions to the equation. Fragmentation or breakage plays a crucial role in the formation of raindrop in clouds and the breakup of asteroids. The fragmentation process can be divided into two types: linear and nonlinear (collision-induced) fragmentation. Linear fragmentation occurs spontaneously or due to some external forces, while nonlinear fragmentation results from collision between particles. Unlike linear fragmentation, nonlinear fragmentation may involve mass transfer between colliding particles, potentially creating particles larger than the original ones. NFE can be further classified based on the presence of mass transfer after collisions: (i) NFE without mass transfer, and (ii) NFE with mass transfer. First, the NFE without mass transfer is examined, where each collisional fragmentation event generates an infinite number of particles, characterized by non-integrable daughter distribution functions. The existence of weak solutions is established for collision kernels exhibiting at most quadratic growth. It is shown that weak solutions conserve mass for all time when the collision kernels have at least linear growth. In contrast, when the collision kernels have sublinear growth, mass conservation holds only for a finite time interval. The analysis relies on weak L1 compactness methods. In both scenarios, uniqueness is ensured under additional constraints on the initial condition. Furthermore, it is shown that mass-conserving weak solutions do not exist for certain collision kernels and a specific class of non-integrable breakage functions defined by a power law.