This presentation will cover two areas of my work: one under the direction of Dr. Luis Caffarelli and one under the direction of Dr. Irene Gamba. In the first part, I will introduce the Grad-Mercier equation in plasma physics, will go over the currently known results, and will present new results of my collaboration with Dr. Luis Caffarelli, Daniel Restrepo, and Ignacio Tomasetti on uniqueness of solutions and free boundary analysis. In the second part, I will present a system of Boltzmann equations for a mixture of gases, will cover existing results on the gain of integrability and propagation of norms for this system, and will explain a potential approach being developed under the direction of Dr. Irene Gamba to show the propagation of norms for the derivatives of the system of Boltzmann equations for a mixture of gases.