Luigi De Rosa

University of Basel

Title: Intermittency and Minkowski content in turbulence

Abstract: 

In 1941 Kolmogorov theorized that all p-th moments of increments of the velocity in a turbulent flow have 1/3 has a universal regularity exponent. However, downward deviations from K41 prediction are experimentally observed. This phenomenon is nowadays known as "intermittency", which theoretical physicists, starting from Landau, linked to the spottiness of the region where the dissipation is supported. We propose a couple of Minkowski-type notions of dimensions, one Eulerian and one Lagrangian, which lay down a setup to make Landau's objection quantitative. The approach is quite geometrical and it is in fact part of a more general picture in which most of the PDEs in fluid dynamics fall.

Enrico Savi

Università di Trento

Title: On the embeddings of compact manifolds

Abstract: 

Barbara Nelli 

Università degli Studi dell'Aquila

Title: The Jenkins-Serrin Theorem reloaded

Abstract: 

Jenkins and Serrin in the sixties proved a famous  theorem   about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical  results, we show how to solve the Jenkins-Serrin problem in a 3-manifold with a Killing vector field. This is a joint work with A. Del Prete and J. M. Manzano.

Emanuela Radici

Università degli Studi dell'Aquila

Title: On curvature and Five Gradients Inequality on Manifolds

Abstract: 

Introduced almost ten years ago, the five gradients inequality has been used to provide estimates on Sobolev norms of minimizers involving the Wasserstein distance. In conjunction with the JKO scheme, this inequality can grant compactness for the minimizing movement scheme. We investigate the geometric and functional meaning of the five gradients inequality in two generalizations. In the setting of Lie groups the proof naturally suggests that it is a second order optimality condition for the Kantorovich potentials, while in general compact Riemannian manifolds the curvature plays a role. This is a joint work with Simone Di Marino and Simone Murro.

Enrico Savi

University of Nice

Title:  In 1936 Whitney proved that every compact smooth manifold $M$ of dimension $d$ is diffeomorphic to a real analytic submanifold of $\mathbb{R}^{2d+1}$. It follows that $M$ can be described both globally and locally as the set of solutions of real analytic equations in some Euclidean space. A natural problem then arises: Is it possible to further simplify the description of $M$ by using polynomial equations with real or even rational coefficients? If the coefficients are real, the answer is affirmative thanks to the Nash-Tognoli theorem. We will discuss this problem, and its relative version, in the case of rational coefficients.

Stefano Spirito

Università degli Studi dell'Aquila

Title: On the inviscid limit for 2D Incompressible Fluid

Abstract:

We review some recent results concerning the inviscid limit for the 2D Euler equations with irregular vorticity. In particular, by using techniques from the theory of transport equations with non smooth vector fields, we show that solutions of the incompressible 2D Euler equations obtained from the ones of the 2D incompressible Navier-Stokes equations via the vanishing viscosity limit satisfy a representation formula in terms of the flow of the velocity and that the strong convergence of the vorticity holds. Moreover, we also prove a rate of convergence. The talk is based on results obtained in collaboration with Gianluca Crippa (Univ. Basel) and Gennaro Ciampa (Università Milano Statale).

Emanuele Tasso

TU Wien

Title: Rectifiability of a class of Integralgeometric measures and applications

Abstract: 

In his textbook 'Geometric Measure Theory', Federer proposed the following problem: (Q) Is the restriction of the m-dimensional Integralgeometric measure to a finite set a m-rectifiable measure?

With this motivation, after a brief introduction to the Integralgeometric measure, I will discuss how rectifiability issues in the spirit of (Q) play an important role in some variational problems. As a sort of unifying theory, I will then introduce a novel class of measures in the euclidean space based upon the idea of slicing. The central result of this talk will follow, which is a sufficient condition for rectifiability in the above class. Two main applications will be shown: the solution to Federer's problem, as well as a novel rectifiability criterion for Radon measures via slicing, the latter being reminiscent of White's rectifiable slices theorem for flat chains. If time permits, I will discuss how to extend the main result to the Riemannian case by means of the notion of transversal family of maps. In the very last part of the talk I will propose some related open problems.