Organized by Institute of Mathematics of UGR (IMAG) in collaboration with Seminario de Ecuaciones Diferenciales de la UGR (SDE).

ABSTRACTS

Claudia García: Patterns and equilibria in incompressible fluids.

The motion of a uniform incompressible fluid is described by the Navier-Stokes equations and, in its inviscid regime, via the Euler equations. In the two-dimensional case, the Euler equations in the vorticity formulation contain many interesting relative equilibria: stationary, rotating and translation solutions. Bifurcation theory arises naturally in the study of many PDE’s, which can be characterized by an implicit equation of the form F(λ,x) = 0 (1), where λ ∈ R and x belongs to some infinite-dimensional Banach space. In this talk, we will take advantage of this theory to review the existence of different kind of solutions: V-states, non uniform rotating vortices or Karman Vortex Street type of solutions, among others. All those simplified dynamics are governed by a nonlinear and nonlocal equation of type (1).

Joan Mateu: On the analyticity of the trajectories of the particles in the patch problem for some active scalar equations.

Let Ω be a bounded domain in Rn whose boundary is C1,γ for γ ∈ (0,1). Consider 2D Euler equation for the vorticity or the n-D aggregation equation in the case of the initial condition being a positive multiple of the characteristic function of Ω. In this talk we discuss on global in time analyticity of the flow generated by the velocity field which propagates the vorticity or density solution respectively. These results are obtained from a detailed study of the Beurling or Riesz transform, that represents derivatives of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.

Zineb Hassainia: On the desingularization of time-periodic vortex motion for the planar Euler equation.

In this talk, I will discuss vortex dynamics in the planar Euler equations, focusing on two key aspects. First, I will present a rigorous derivation of leapfrogging quartets of concentrated vortex patches near singular time-periodic relative equilibria of the point vortex system, using KAM theory. In the second part, I will show how to extend these techniques to desingularize time-periodic vortex orbits when the Euler equation is set in a generic bounded simply connected domain. Specifically, we can prove that for a single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches.