Least action principles with applications to gradient flows and kinetic equations
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This thesis introduces a variational formulation for a family of kinetic reaction-diffusion 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. We use this class to build solutions to reaction-diffusion equations with drift subject to general Dirichlet boundary condition via an extension of De Giorgi's interpolation method for the entropy functional. In 2010, Alessio Figalli and Nicola Gigli introduced a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in their work. The costs we introduce allow the drift and the boundary condition to be decoupled. 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 diameter of the interval can be made arbitrarily small by choosing the force sufficiently large.