Research Interests

Description of my Research

The partial differential equations (PDEs) which appear in mathematical fluid mechanics describe the time evolution of a fluid. One expects that repeating a particular experiment with a real fluid several times always leads to the same observation. This fact should be recovered in terms of uniqueness of the solution to the underlying mathematical model (i.e. the corresponding PDE). However, for many fluid models one can construct infinitely many solutions to given initial datum using a method called "convex integration". This non-uniqueness questions the solution concept which is used here (so-called weak entropy solutions).

Moreover, with the help of convex integration one was able to prove Onsager's conjecture, which comes from turbulence theory and shows how regular turbulent flows can be. This indicates that there is a relation between non-uniqueness and turbulence.

The goal of my research is to find a suitable solution concept, in particular in view of turbulent flows. Can we retrieve the notion of a weak solution by imposing an additional criterion which singles out one solution among the infinitely many? Or do we have to consider a generalized notion of solution instead (e.g. measure-valued solutions)? Should we expect that turbulent solutions are unique? Why does the concept of weak entropy solutions allow for the existence of many solutions in multiple space dimensions, whereas it leads to satisfying existence and uniqueness results in one dimension?

Furthermore, my research shall enhance the mathematical understanding of the physical phenomenon of turbulence. Which properties do turbulent solutions have (e.g. in view of regularity)? Do these properties depend on the model under consideration?