Doubly nonlinear evolution equations

In this project we consider doubly nonlinear evolution equations which can be seen as a combination of the parabolic p-Laplace equation and the Porous Medium Equation. This type of partial differential equations posses a wide spectrum of applications, for instance in fluid dynamics, soil science and filtration. Despite their importance in mathematics and applications, there are still many open questions in the field and the mathematical understanding of these equations is only at the beginning. The aim of this project is to create new insight into both, the existence and regularity theory for doubly nonlinear evolution equations. Although the project is mainly concerned with theoretical analytical questions, the considered equations are motivated by applications and there are points of contact with numerics and engineering. The project consists of two parts. The first part is concerned with the existence theory of doubly nonlinear evolution problems. Here, we introduce a variational viewpoint of the underlying equations. This allows to develop a purely variational existence theory for doubly nonlinear evolution equations via a non-linear version of the Minimizing Movements (Implicit Euler) method. This method will be applied in various settings like obstacle problems or a fast diffusion variant of the minimal surface equation. Furthermore, we will investigate uniqueness and approximation properties of solutions. The solutions obtained so far are only generalized solutions in certain Sobolev spaces. Therefore, in the second part of the project, we will investigate their regularity properties. In particular, we are interested in Hölder regularity and so-called higher integrability results. By this we mean a small improvement of the integrability of the spatial gradient of solutions. Such properties are not only interesting by themselves. They are usually important ingredients in the proof of other regularity results such as partial regularity or Calderòn-Zygmund estimates.

In this project we consider doubly nonlinear evolution equations which can be seen as a combination of the parabolic p-Laplace equation and the Porous Medium Equation. This type of partial differential equations posses a wide spectrum of applications, for instance in fluid dynamics, soil science and filtration. Despite their importance in mathematics and applications, there are still many open questions in the field and the mathematical understanding of these equations is only at the beginning. We succeeded to create new insight into both, the existence and regularity theory for doubly nonlinear evolution equations. Although the project is mainly concerned with theoretical analytical questions, the considered equations are motivated by applications in physics and engineering and there are points of contact with numerics. The project consists of two parts. The first part is concerned with the existence and uniqueness theory of doubly nonlinear evolution problems. Here, we introduce a variational viewpoint of the underlying equations. This allows to develop a purely variational existence theory for doubly nonlinear evolution equations via a non-linear version of the Minimizing Movements (Implicit Euler) method. This method was applied in various settings like the obstacle problem. Furthermore, we proved a comparison principle which in turn allowed to gain new insight into uniqueness of solutions. The solutions obtained so far are only generalized solutions in certain Sobolev spaces. Therefore, in the second part of the project, we investigate their regularity properties. In particular, we proved Hölder regularity of the solution and its spatial gradient and so-called higher integrability results. By this we mean a small improvement of the integrability of the spatial gradient of solutions. Such properties are not only interesting by themselves. They are usually important ingredients in the proof of other regularity results such as partial regularity or Calderòn-Zygmund estimates.