Institut Camille Jordan

We consider the surface quasi-geostrophic equation with
critical dissipation and dispersive forcing set on the vertical strip $\Omega=
[-1, 1] \times \mathbb{R}^2 $, with homogeneous Dirichlet boundary conditions for the surface temperature. Similar models for the quasi-geostrophic ocean circulation model of potential vorticity have been treated, where the presence of horizontal boundary layers serves to represent the western intensification of boundary currents.

Our aim is to display this phenomenon by constructing a boundary layer approximation which converges to the global weak solutions of the system, in the limit of large dispersive forcing. We adapt the Strichartz estimates from the full space to the strip and use the dispersive effects to control the nonlinearity. As in the case of the full space this also allows us to show that the asymptotic behavior of solutions is determined by the linear part of the system, that is, we obtain the stabilization effect by Rossby wave propagation. The convergence of the approximations is shown in the energy norm.

Work with Gabriela Planas (Unicamp, Campinas, Brazil)

Since Kuramoto proposed a model of coupled oscillators, the study of synchronization has pulled the attention from different point of views: biology, chemistry, neuroscience, etc. Such a phenomenon is observed very often in biological systems like the flashing of fireflies, the beating of heart cells and the synaptic firing of neurons in the brain. In this talk we review the state of the art for the Kuramoto model and its associated kinetic counterpart, namely, the Kuramoto—Sakaguchi equation.

The heart of our talk will be to introduce new quantitative estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto—Sakaguchi equation departing from generic initial data in a large coupling strength regime. Although some previous attempts have been presented for oscillators confined to a half circle, this is the first result in the literature that covers generic initial data, to the best of our knowledge. Notice that the lack of convexity of the problem and the presence of heterogeneities prevent us from using classical methods for gradient flows. Then, the key step will be to introduce a new transportation distance that endows the system with the appropriate structure. In particular, we show new generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by Otto and Villani, that quantity convergence to equilibrium.