# Research

The main focus of this group is on rigorous applied mathematics, mainly involving partial diﬀerential equations and the calculus of variations. We bridge several mathematical sub-disciplines by applying methods originally developed for diﬀerential geometry (Gromov) to ﬂuid dynamics and nonlinear elasticity.

Our work impacts long-held beliefs on so-called weak solutions to the fundamental equations of continuum mechanics, like Euler’s equation for incompressible or compressible ﬂuids. Previously, the notion of weak solutions was seen as a convenient relaxation of classical solutions, closer to the physical conservation laws of mass and momentum. Our work established that weak solutions are under-determined to a dramatic degree: Any closure model for the Reynold’s stress can be approximated by exact weak solutions to Euler’s equation.

We applied this theory to underpin Onsager’s vision that the relevant dissipative – and thus weak – solutions of Euler’s equation have a very speciﬁc roughness, namely a third of a spatial derivative. We proved that there is indeed an abundance of strictly dissipative solutions to Euler’s equation of this borderline behavior. We did so through an ingenious multi-scale construction based on an elementary ﬂow pattern. On the other hand, we drew an analogy to an isometric embedding problem from classical diﬀerential geometry with its transition from rigidity (Herglotz) to ﬂoppiness (Nash). In fact, as opposed to ﬂuid dynamics, the critical exponent in this paper crumpling problem is unknown, and we provide promising contributions to narrowing down the range.

We have validated this approach in diverse applications like the microstructure generated by the density-driven instability in a two-phase porous medium ﬂow, or the microstructure emerging from a vortex sheet. Recently, we have ventured into magneto-hydrodynamics with its conservation of magnetic helicity alongside the dissipation of energy. Our research can be seen as a multi-pronged contribution to a rational mechanics of oscillations and evolving microstructure (Tartar).

## Positions

There are **open positions for Max Planck Research Group Leaders (W2)** in our group at the Max Planck Institute:

- See more information about the position and how to apply at our career page.

There are **open positions for PhD students and postdocs** in our group at the Max Planck Institute:

- PhD Stipends: see International Max Planck Research School "Mathematics in the Sciences"
- Postdoc fellowships and visiting positions: see Advertisement on the Institute's webpage
- Applications by candidates of outstanding qualification and a good fit with our research goals will always be considered