I will present some recent results obtained in collaboration with V. Banica and F. de la Hoz on the evolution of vortex filaments according to the so called Localized Induction Approximation (LIA). This approximation is given by a non-linear geometric partial differential equation, that is known under the name of the Vortex Filament Equation (VFE). The aim of the talk is threefold. First, I will recall the Talbot effect of linear optics. Secondly, I will give some explicit solutions of VFE where this Talbot effect is also present. Finally, I will consider some questions concerning the transfer of energy and momentum for these explicit solutions.

# Past Partial Differential Equations Seminar

In a given ambient Riemannian manifold with boundary, geometric objects of particular interest are those properly embedded submanifolds that are critical points of the volume functional, when allowed variations are only those that preserve (but not necessarily fix) the ambient boundary. This variational condition translates into a quite nice geometric condition, namely, minimality and orthogonal intersection with the ambient boundary. Even when the ambient manifold is simply a ball in the Euclidean space, the theory of these objects is very rich and interesting. We would like to discuss several aspects of the theory, including our own contributions to the subject on topics such as: classification results, index estimates and compactness (joint works with different groups of collaborators - I. Nunes, A. Carlotto, B. Sharp, R. Buzano - will be appropriately mentioned).

We discuss mathematical questions that play a fundamental role in quantitative analysis of incompressible viscous fluids and other incompressible media. Reliable verification of the quality of approximate solutions requires explicit and computable estimates of the distance to the corresponding generalized solution. In the context of this problem, one of the most essential questions is how to estimate the distance (measured in terms of the gradient norm) to the set of divergence free fields. It is closely related to the so-called inf-sup (LBB) condition or stability lemma for the Stokes problem and requires estimates of the LBB constant. We discuss methods of getting computable bounds of the constant and espective estimates of the distance to exact solutions of the Stokes, generalized Oseen, and Navier-Stokes problems.

After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy- Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow. Joint work with F. Cavalletti.

Existence results in large for fully non-linear compressible multi-fluid models are in the mathematical literature in a short supply (if not non-existing). In this talk, we shall recall the main ideas of Lions' proof of the existence of weak solutions to the compressible (mono-fluid) Navier-Stokes equations in the barotropic regime. We shall then eplain how this approach can be adapted to the construction of weak solutions to some simple multi-fluid models. The main tools in the proofs are renormalization techniques for the continuity and transport equations. They will be discussed in more detail.

In this presentation, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to ε. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential uε caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possibly negative) conductivity a surrounded by a dielectric medium, with unit conductivity.

In this joint work with Amandine Aftalion we study the minimisers of an energy functional in two-dimensions describing a rotating two-component condensate. This involves in particular separating a line-energy term and a vortex term which have different orders of magnitude, and requires new estimates for functionals of the Cahn-Hilliard (or Modica-Mortola) type.

We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that

maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This

is joint work with Henrik Matthiesen.

For liquid films with a thickness in the order of 10¹−10³ molecule layers, classical models of continuum mechanics do not always give a precise description of thin-film evolution: While morphologies of film dewetting are captured by thin-film models, discrepancies arise with respect to time-scales of dewetting.

In this talk, we study stochastic thin-film equations. By multiplicative noise inside an additional convective term, these stochastic partial differential equations differ from their deterministic counterparts, which are fourth-order degenerate parabolic. First, we present some numerical simulations which indicate that the aforementioned discrepancies may be overcome under the influence of noise.

In the main part of the talk, we prove existence of almost surely nonnegative martingale solutions. Combining spatial semi-discretization with appropriate stopping time arguments, arbitrary moments of coupled energy/entropy functionals can be controlled.

Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments - in particular on Jakubowski’s generalization of Skorokhod’s theorem - weak convergence methods, and recent tools for martingale convergence.

The results have been obtained in collaboration with K. Mecke and M. Rauscher and with J. Fischer, respectively