Highly nonlinear evolutionary problems

在“高度非线性进化问题”项目中，我们主要关注时相关偏微分方程和最小化性质。我们将讨论三种不同的问题。首先，我们考虑双非线性抛物方程，它出现在几种物理现象的建模中，例如，在等离子体物理或气体或液体通过多孔介质的湍流过滤分析中。这些方程也可用于描述地下水问题、等离子体中的热辐射或粘性流体的运动。本课题的一个主要目的是利用正性展开的方法研究这些方程解的自改进性质。在过去的几年里，人们对在随时间变化的域中形成的偏微分方程相关的问题越来越感兴趣。这部分是由于数学生物学中的许多问题自然地提出了生长领域（例如发展生物体或增殖细胞）或以某种特定方式进化的领域。还有一些经典的工程应用，如流体或气体在管道或管道中设置有限制壁，可以随意置换、移除或引入。目的是展示描述这些现象的进化问题的变分解的存在性结果。该方法将基于DeGeorgi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，在假设增长率不是“太大”的情况下，与此类解决方案相关的抛物线最小值在某种程度上是平滑的。为了证明这一点，我们将使用DeGeorgi类的抛物线版本和Moser迭代的一个变体。

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.