非线性扩散方程解的多尺度方法渐近分析
Análise Assintótica de Soluções de Equações Difusivas Não-Lineares via Métodos de Escalas Múltiplas
AI总结：
本文通过重整化群和均质化方法，研究具有周期系数或时间依赖系数的非线性扩散方程解的渐近行为，并给出数值重整化群方法及相变曲线。
摘要：
在本工作中，我们将描述并应用基于数据缩放和算子重整化的重整化群技术，以及均质化技术（在特定极限下，将周期非均匀介质替换为均匀介质），来研究偏微分方程解的渐近行为。本论文研究的一类问题包括具有周期系数的非线性扩散微分方程，这些方程假设用于模拟非均匀介质中的扩散。我们对这些问题的研究涉及上述两种技术的论证，这在某种意义上建立了它们之间的联系。这里研究的另一类问题包括具有时间依赖系数的非线性扩散方程，这些方程源于对模拟多孔介质中两相流现象的随机微分方程平均解渐近行为的蒙特卡洛模拟。利用重整化群，我们将根据问题的定性行为对其进行分类，并通过数值验证。我们还将研究分离线性与非线性渐近区域的相变曲线。本工作的主要结果由数值重整化群版本给出，其将在论文正文中详细描述。
英文摘要：
In the present work we shall describe and apply the techniques of the Renormalization Group - based in data rescaling and operator renormalizing - and of Homogenization - that substitutes, in a certain limit, a periodically inhomogeneous medium by a homogeneous one - for the study of asymptotic behavior of solutions of PDE's. One class of problems studied in this dissertation consists of nonlinear, diffusive differential equations with periodic coefficients, that supposedly model the diffusion in a heterogeneous medium. Our study of these problems involves arguments originated from the two techniques described above. This makes, in certain sense, a connection between both of them. Another class of problems studied here consists of nonlinear, diffusive equations with time dependent coefficients, originated from montecarlo simulation of the asymptotic behavior of the averaged solutions of stochastic differential equations modeling the phenomenon of two-phase flow in porous media. Using the Renormalization Group we shall classify the problems in accordance with their qualitative behavior. This classification will be verified numerically. We shall also study the curve of phase transition that separates asymptotic regimes predominantly linear and nonlinear. The main results presented in this work are given by a numerical version of the Renormalization Group, which will be described in detail in the body of this dissertation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
