Mathematics Colloquia and Seminars

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Description

We study the evolution of system of $n$ particles $\{(x_i, v_i)\}_{i=1}^n$ in $\R2d$. 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 $\mu$ is a discrete measure concentrated on the set $\{(x_i, v_i)\}_{i=1}^n$. Typically, $\mu(0)$ is a discrete measure approximating an initial $L^\infty$ density and can be chosen randomly. When $d=1$ our result 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 Lebesgue measure. When $\{\nu^n\}_{n=1}^\infty$ converges to a measure concentrated on a special $d$--dimensional sets, we obtain the Vlasov-Monge-Amp\`ere (VMA) system. When, $d=1$ the VMA system coincides with the standard Vlasov-Poisson system.