Alexander Mielke's 65th Birthday

Three examples from geophysical fluid dynamics will showcase mathematical modelling as the "art of judicious simplification": The computational prediction of two seasonal to decadal phenomona, the "quasi-biennial oscillation" (QBO) and the "El Niño Southern Oscillation" (ENSO) became possible only after theoreticians had captured their essential causal structures in convincing reduced mathematical models. With our own research, we aim to similarly untangle the mechanisms behind the "rapid intensification" (RI) of tropical storms during their transition to hurricane strength.

Singular stochastic PDE are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. The guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow the approach of mathematical physics, and of Hairer's regularity structures, which however we re-interpret as providing an informal parameterization of the infinite-dimensional nonlinear solution manifold.

We systematically follow this more geometric/analytic than combinatorial point-of-view: Instead of appealing to an expansion indexed by trees, we consider all partial derivatives w.r.t. the "constitutive" function defining the nonlinearity. Instead of a Gaussian calculus guided by Feynman diagrams arising from pairing nodes of two trees, we consider derivatives w.r.t. the noise, i.e. Malliavin derivatives. We interpret the Malliavin derivative of the parameterization as an approximate tangent vector to the solution manifold, which yields a sparse representation in terms of the parameterization itself, and paves the way for its stochastic estimate. Ultimately, this gives a characterization of the solution manifold that is oblivious to the divergent counter terms.

This is work with L. Broux, P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.

The mathematical analysis of rate-independent systems has sparked intensive research over the last two decades, both because they are ubiquitous in mechanics, and due to their intrinsic nonsmoothness, which poses several challenges. The work by Alexander Mielke has set a cornerstone in the field, starting with the weak solvability notion of Energetic solution. The Energetic concept has become paradigmatic for rate-independent evolution: one of its fingerprints is its variational character, which paves the way for the usage of the wide toolbox of Calculus of Variations in the analysis of specific systems. And, again, the interplay between rate-independent systems and the variational theory for gradient flows has inspired yet another concept, that of Balanced Viscosity solutions. A twitch upon the thread leads to Alexander's work on discrete systems, to EDP convergence... and to several intriguing open perspectives.