Rigorous Results In Fluid And Kinetic Models

In the following, we will consider two different physical systems and their respective PDE models. In the first chapter, we prove time decay of solutions to the Muskat equation, which describes a fluid interface between two incompressible, immiscible fluids with different densities. In \cite{JEMS} and \cite{CCGRPS}, the authors introduce the norms $$ |f|{s}\eqdef \int{\mathbb{R}^{2}} |\xi|^{s}|\hat{f}(\xi)| \ d\xi $$ in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data $f_{0}\in H^{l}(\mathbb{R}^{2})$ for some $l\geq 3$ such that $|f_{0}|{1} < k{0}$ for a constant $k_{0} \approx 1/5$, we prove uniform in time bounds of $|f|{s}(t)$ for $-2 < s < l-1$ and assuming $|f{0}|{\nu} < \infty$ we prove time decay estimates of the form $|f|{s}(t) \lesssim (1+t)^{-s+\nu}$ for $0 \leq s \leq l-1$ and $-2 \leq \nu < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We prove analogous results in 2D. In the remaining chapters, we consider sufficient conditions, called continuation criteria, for global existence and uniqueness of classical solutions to the three-dimensional relativistic Vlasov-Maxwell system. In the compact momentum support setting, we prove that $|p_{0}^{\frac{18}{5r} - 1+\beta}f|{L^{\infty}{t}L^{r}{x}L^{1}{p}} \lesssim 1$ where $1\leq r \leq 2$ and $\beta >0$ is arbitrarily small, is a continuation criteria. The previously best known continuation criteria in the compact setting is $|p_{0}^{\frac{4}{r} - 1+\beta}f|{L^{\infty}{t}L^{r}{x}L^{1}{p}} \lesssim 1$, where $1\leq r < \infty$ and $\beta >0$ is arbitrarily small, due to Kunze \cite{Kunze}. Our continuation criteria is an improvement in the $1\leq r \leq 2$ range. We also consider sufficient conditions for a global existence result to the three-dimensional relativistic Vlasov-Maxwell system without compact support in momentum space. In Luk-Strain \cite{Luk-Strain}, it was shown that $|p_{0}^{\theta}f|{L^{1}{x}L^{1}_{p}} \lesssim 1$ is a continuation criteria for the relativistic Vlasov-Maxwell system without compact support in momentum space for $\theta > 5$. We improve this result to $\theta > 3$. We also build on another result by Luk-Strain in \cite{L-S}, in which the authors proved the existence of a global classical solution in the compact regime if there exists a fixed two-dimensional plane on which the momentum support of the particle density remains bounded. We prove well-posedness even if the plane varies continuously in time.