Archive for Rational Mechanics and Analysis

, Volume 137, Issue 2, pp 99-134

First online:

An Integro-Differential Equation Modelling a Newtonian Dynamics and Its Scaling Limit

  • Karl OelschlägerAffiliated withInstitut für Angewandte Mathematik Universität Heidelberg Im Neuenheimer Feld 294 D - 69120 Heidelberg

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We consider an integro-differential equation describing a Newtonian dynamics with long-range interaction for a continuous distribution of mass in R. First, we deduce unique existence and regularity properties of its solution locally in time, and then we investigate a scaling limit. As limit dynamics a nonlinear wave equation is determined. Technically, we rely on the connection of the Newtonian dynamics to a system of an integro-differential equation and a partial differential equation. Basic for our considerations is the study of the regularity properties of the solution of that system. For that purpose we exploit its similarity to a certain strongly hyperbolic system of partial differential equations.