We study the evolution of a system of n particles in a 2d-dimensional euclidean space. That system is a conservative system with a Hamiltonian of the form H(h)=W(h,kn), where W is the Wasserstein distance and m is a discrete measure. Typically, ho is a discrete measure approximating an initial essentially bounded 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 Lebesgue measure. When kn converges to a measure concentrated on a special d-dimensional sets, we obtain the Vlasov-Monge-Ampère (VMA) system. When d=1, the VMA system coincides with the standard Vlasov-Poisson system.
The Semigeostrophic Equations Discretized in Reference and Dual Variables
PISANTE, Giovanni;
2007
Abstract
We study the evolution of a system of n particles in a 2d-dimensional euclidean space. That system is a conservative system with a Hamiltonian of the form H(h)=W(h,kn), where W is the Wasserstein distance and m is a discrete measure. Typically, ho is a discrete measure approximating an initial essentially bounded 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 Lebesgue measure. When kn converges to a measure concentrated on a special d-dimensional sets, we obtain the Vlasov-Monge-Ampère (VMA) system. When d=1, the VMA system coincides with the standard Vlasov-Poisson system.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
