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Highly nonlinear evolutionary problems

高度非线性进化问题

基本信息

批准号：
396311282
负责人：
Dr. Thomas Singer
金额：
--
依托单位：
Department Mathematik
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/396311282?language=en
关键词：
Highly nonlinear evolutionary problems

项目摘要

In the project “Highly nonlinear evolutionary problems”, we are mainly concerned with time dependent partial differential equations and minimizing properties. We will treat three different kinds of problems. First, we consider doubly nonlinear parabolic equations, which appear in the modelling of several physical phenomena as, for instance, in plasma physics or the analysis of turbulent filtration of a gas or a liquid through porous media. Those equations also find application in the characterization of ground water problems, heat radiation in plasmas, or the motion of viscous fluids. One main goal of this project is to investigate a self-improving property of solutions of these equations by using the method of Expansion of Positivity.In the last years, the interest in problems that are related with partial differential equations formulated in domains that change in time grew. This is partly due to the fact that a number of problems in mathematical biology are naturally posed on growing domains (e.g. developing organisms or proliferating cells) or domains that evolve in some particular way. There are also some classical engineering applications like fluids or gases in settings as channels or pipes with confining walls that may be displaced, removed or brought in at will. The aim is to show existence results for variational solutions to evolutionary problems that describe these phenomena. The approach will be based on DeGeorgi’s method of minimizing movements.Finally, we will consider functionals with an exponential growth rate. The purpose is to show that parabolic minimizers that are associated to such solutions are in some way smooth, assuming that the growth rate is not “too large”. To prove this, we will make use of a parabolic version of DeGeorgi classes and a variant of the Moser iteration.

在“高度非线性演化问题”项目中，我们主要研究了时间依赖的偏微分方程及其极小化性质。我们将讨论三种不同类型的问题。首先，我们考虑双非线性抛物方程，它出现在几个物理现象的建模，例如，在等离子体物理或分析的气体或液体通过多孔介质的湍流过滤。这些方程也可用于描述地下水问题、等离子体热辐射或粘性流体运动。本项目的主要目的之一是利用正性展开法研究这些方程的解的自改进性质。近年来，人们对在随时间变化的区域中形成的偏微分方程相关问题的兴趣不断增长。这部分是由于数学生物学中的许多问题自然地在生长域（例如发育中的生物体或增殖细胞）或以某种特定方式进化的域上提出。也有一些经典的工程应用，如流体或气体在设置为通道或管道与限制壁，可以取代，删除或带来的意愿。其目的是显示存在的结果变分解决方案的进化问题，描述这些现象。该方法将基于DeGeorgi的最小化运动的方法。最后，我们将考虑具有指数增长率的泛函。其目的是表明，抛物线极小，与这种解决方案是在某种程度上顺利，假设的增长率是不是“太大”。为了证明这一点，我们将使用抛物线版本的DeGeorgi类和莫泽迭代的变体。