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“高度非线性进化问题”中，我们将对待不同的问题，我们认为非线性抛物线方程出现在几种现象中。气体或液体的培养基。统一的扩展是去年的，对与域中的部分方程相关的问题的兴趣会变化的域，部分原因是数学生物学自然是在生长域SMS或增殖细胞上的）或在某些方面，有些经典的工程应用程序，例如流体或气体，作为通道或管道墙，可能会流离失所。我们将以指数的增长速度建设。