Abstract:
In this talk, we introduce a variational formulation for a family of kinetic equations and their connection to Lagrangian dynamical systems. Such a formulation uses a new class of transportation costs between positive measures, and it generalizes the notion of gradient flows. Additionally, we use this variational formulation to obtain well-posedness, stability, and convergence to equilibrium for the homogeneous Vicsek model and to show the emergence of phase concentration for the Kuramoto Sakaguchi equation subject to a strong coupling force. Provided this coupling force is sufficiently large, we show that there exists a time-dependent interval such that the oscillator's probability density converges to zero uniformly in its complement. The length of this interval is quantified as a function of the coupling force and the diameter of the support of the natural frequency distribution. By doing this, we show that the width of the interval can be made arbitrarily small by choosing the force to be sufficiently large. |