Rigorous Results In Fluid And Kinetic Models

Abstract

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 f0Hl(R2)f_{0}\in H^{l}(\mathbb{R}^{2}) for some l3l\geq 3 such that f013˘ck0\|f_{0}\|_{1} \u3c k_{0} for a constant k01/5k_{0} \approx 1/5, we prove uniform in time bounds of fs(t)\|f\|_{s}(t) for 23˘cs3˘cl1-2 \u3c s \u3c l-1 and assuming f0ν3˘c\|f_{0}\|_{\nu} \u3c \infty we prove time decay estimates of the form fs(t)(1+t)s+ν\|f\|_{s}(t) \lesssim (1+t)^{-s+\nu} for 0sl10 \leq s \leq l-1 and 2ν3˘cs-2 \leq \nu \u3c 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 p0185r1+βfLtLxrLp11\|p_{0}^{\frac{18}{5r} - 1+\beta}f\|_{L^{\infty}_{t}L^{r}_{x}L^{1}_{p}} \lesssim 1 where 1r21\leq r \leq 2 and β3˘e0\beta \u3e0 is arbitrarily small, is a continuation criteria. The previously best known continuation criteria in the compact setting is p04r1+βfLtLxrLp11\|p_{0}^{\frac{4}{r} - 1+\beta}f\|_{L^{\infty}_{t}L^{r}_{x}L^{1}_{p}} \lesssim 1, where 1r3˘c1\leq r \u3c \infty and β3˘e0\beta \u3e0 is arbitrarily small, due to Kunze \cite{Kunze}. Our continuation criteria is an improvement in the 1r21\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 p0θfLx1Lp11\|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 θ3˘e5\theta \u3e 5. We improve this result to θ3˘e3\theta \u3e 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

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