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