Browsing by Subject "weak*-topology"

The Smoluchowski coagulation equation is considered to be one of the most fundamental equations of the classical description of matter alongside with the Boltzman, Navier-Stokes and Euler equations. It has applications from physical chemistry to astronomy. In this thesis, a new existence result of measure valued solutions to the coagulation equation is proven. The proven existence result is stronger and more general than a previously claimed result. The proven result holds for a generic class of coagulation kernels, including various kernels used in applications. The coagulation equation models binary coagulation of objects characterized by a strictly positive real number called size, which often represents mass or volume in applications. In binary coagulation, two objects can merge together with a rate characterized by the so-called coagulation kernel. Time evolution of the size distribution is given by the coagulation equation. Traditionally the coagulation equation has two forms, discrete and continuous, which are referring to whether the objects sizes can take discrete or continuous values. A similar existence result to the one proven in this thesis has been obtained for the continuous coagulation equation, while the discrete coagulation equation is often favored in applications. Being able to study both discrete and continuous systems and their mixtures at the same time has motivated the study of measure valued solutions to the coagulation equation. After motivating the existence result proven in this thesis, its proof is organized into four Steps described at the end of the introduction. The needed mathematical tools and their connection to the four Steps are presented in chapter 2. The precise mathematical statement of the existence result is given in chapter 3 together with Step 1, where the coagulation equation will be regularized using a parameter ε ∈ (0, 1) into a more manageable regularized coagulation equation. Step 2 is done in chapter 4 and it consists of proving existence and uniqueness of a solution f_ε for each regularized coagulation equation. Step 3 and Step 4 are done in chapter 5. In Step 3, it will be proven that the regularized solutions {f_ε} have a converging subsequence in the topology of uniform convergence on compact sets. Step 4 finishes the existence proof by verifying that the subsequence’s limit satisfies the original coagulation equation. Possible improvements and future work are outlined in chapter 6.