Seminários

We study a fractional Yamabe-type equation on compact Riemannian manifolds equipped with an isoparametric function. This symmetry reduces the nonlocal problem to a one-dimensional weighted equation on the space of orbits. After setting up the fractional Sobolev framework and the isoparametric reduction, we show that variational methods, namely the Symmetric Mountain Pass Theorem, produce infinitely many solutions to the reduced problem. Time permitting, we will also outline how the same reduction applies to bifurcation phenomena for this family of equations, identifying the bifurcation points through linearization around the constant solution.

In this talk, I will discuss the growth rate of Reeb orbits with respect to their periods on certain contact manifolds. We focus on fiberwise star-shaped hypersurfaces in the cotangent bundle of a closed manifold. This class of dynamical systems includes, for example, geodesic flows. I will explain how the string topology of the base manifold implies that the number of Reeb orbits with period at most grows at least like the prime numbers, that is, on the order of . The topological condition is the existence of a certain non-nilpotent homology class of the free loop space (with respect to the Chas–Sullivan product). This is joint work with João Pering.

Despite the intrinsic complexity of turbulent flows, large-scale coherent structures are present in turbulence, and are related to quantities of interest, such as aerodynamic drag and sound radiation. This presentation shows methods to model such coherent structures, and approaches to control flows in order to reduce drag or noise radiation. Linear models will be considered first, with coherent structures modelled as dominant responses from the linearised Navier-Stokes system. Such dominant modes compare favourably with coherent structures educed from experimental or numerical data, and allow the proposition of changes to the system aiming at drag or noise reduction. Non-linear reduced-order models (ROMs) are then obtained by the Galerkin projection of the full governing equations onto modes obtained from the linearised system. Dynamical systems so obtained typically have dimensions of O(10) to O(10^3) degrees of freedom, and quantitative agreement with reference statistics is obtained for a number of canonical flow configurations. State estimation from a few sensors is explored using an extended Kálmán filter coupled with ROMs, reproducing accurately numerical results. Finally, a ROM for turbulent Couette flow is used to devise a control method aiming at turbulence suppression. The obtained strategy is seen to relaminarise turbulence in direct numerical simulations, showing that ROMs so obtained have promising applications in optimisation and control.