The Semigeostrophic Equations Discretized in Reference and Dual Variables

Abstract

We study the evolution of a system of n particles ${\{(x_i, v_i)\}_{i=1}^{n}}$ in ${\mathbb{R}^{2d}}$ . That system is a conservative system with a Hamiltonian of the form ${H[\mu]=W^{2}_{2}(\mu, \nu^{n})}$ , where W ₂ is the Wasserstein distance and μ is a discrete measure concentrated on the set ${\{(x_i, v_i)\}_{i=1}^{n}}$ . Typically, μ(0) is a discrete measure approximating an initial L ^∞ density and can be chosen randomly. When d  =  1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to the Lebesgue measure. When ${\{\nu^n\}_{n=1}^\infty}$ converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system.