## Titles and Abstracts:

### Franco Flandoli (SNS Pisa):Effect of transport noise on PDEs

Abstract. Linear PDEs of transport type are regularized by the addition of an extra
transport term of stochastic type; this is a phenomenon discovered around 2010 and
consolidated by different techniques and on different examples. However, the effect on
nonlinear PDEs is much less clear. Two results for point vortex solutions and point charge
solutions of 2D Euler equations and 1D Vlasov-Poisson equations respectively, indicate
that a rich noise has to be considered, opposite to the linear case where a simple
space-independent noise suffices to regularize. Some confirmations that such rich noise
may have regularizing properties on nonlinear models came for Leray alpha model and dyadic
models of turbulence. We now have new insight into the case 3D Navier-Stokes equations,
that will be explained in the talk. The results mentioned above, the classical and the
new ones, have been obtained by several authors including M. Gubinelli, E. Priola,
D. Barbato, L. Galeati, D. Luo and myself.
### Philipp Harms (Univ. Freiburg): Infinite-dimensional Riemannian geometry as a mathematical basis for
shape analysis and fluid dynamics

Shape analysis and fluid dynamics can be developed on the same
mathematical basis, namely Riemannian geometry on nonlinear mapping
spaces. In shape analysis the geodesic boundary value problem provides
a similarity metric and point correspondences between shapes. In fluid
dynamics the geodesic initial value problem describes the dynamics of
the fluid. I will give an overview of some of these connections and
present some recent results on local well-posedness of a large class
of variational PDEs associated to fractional Laplace operators.
### Camilla Nobili (Univ. Hamburg): Rigorous bounds on
Rayleigh-Bénard convection at finite Prandtl number

We are interested in thermal convection as described by the Rayleigh-Bénard convection
model. In this model the Navier-Stokes equations for the (divergence-free) velocity u
with no-slip boundary conditions are coupled to an advection-diffusion equation for the
temperature T with inhomogeneous Dirichlet boundary conditions. The problem of
understanding the (average) upward-heat-transport properties is of great interest for the
applications and challenging for the rigorous analysis. We show how the PDE theory (in
particular, regularity analysis) can contribute to the understanding of the scaling
regimes for the heat transport. After reviewing the theory of Constantin & Doering 1999 we
will present some recent results and discuss new challenges.
### Enrico Valdinoci (Univ. Perth): Nonlocal minimal graphs in the plane are generically sticky

We discuss some recent boundary regularity results for nonlocal minimal surfaces in the
plane. In particular, we show that nonlocal minimal graphs in the plane exhibit
generically stickiness effects and boundary discontinuities. More precisely, if a nonlocal
minimal graph in a slab is continuous up to the boundary, then arbitrarily small
perturbations of the far-away data necessarily produce boundary discontinuities. Hence,
either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation
of the prescribed conditions produces boundary discontinuities. The proof relies on a
sliding method combined with a fine boundary regularity analysis, based on a
discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs
are either discontinuous at the boundary or their derivative is Hölder continuous up to
the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal
graphs in the plane "jumps" from discontinuous to differentiable, with no intermediate
possibilities allowed. In particular, we deduce that the nonlocal curvature equation is
always satisfied up to the boundary. As a byproduct of our analysis, one describes the
"switch" between the regime of continuous (and hence differentiable) nonlocal minimal
graphs to that of discontinuous (and hence with differentiable inverse) ones. These
results have been obtained in collaboration with Serena Dipierro and Ovidiu Savin.
### Ruijun Wu (SNS Pisa): Super Liouville equations on closed Riemann surfaces

The super Liouville equations come from the supersymmetric extension of the
classical Liouville field theory. It couples the Liouville equation with a critical Dirac
equation. Since the Dirac operators have infinite negative part in the spectrum, the
problems are usually variationally nontrivial. We will talk about some recent existence
results for such equations, obtained via variational methods. This is a joint work with
A. Jevniker and A. Malchiodi.

Last modified: Wed Jan 8 10:04:41 CET 2020