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A variational representation of weak solutions for the pressureless Euler-Poisson equations

We derive an explicit formula for global weak solutions of the one dimensional system of pressure-less Euler-Poisson equations. Our variational formulation is an extension of the well-known formula for entropy solutions of the scalar inviscid Burgers' equation: since the characteristics of the Euler-Poisson equations are parabolas, the representation of their weak solution takes the form of a "quadratic" version of the celebrated Lax-Oleinik variational formula. Three cases are considered. (i) The variational formula recovers the "sticky particle" solution in the attractive case; (ii) It represents a repulsive solution which is different than the one obtained by the sticky particle construction; and (iii) the result is further extended to the multi-dimensional Euler-Poisson system with radial symmetry.