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We prove the convergence of a discontinuous Galerkin method approximating the 2-D incompressible Euler equations with discontinuous initial vorticity:$omega_0 in L^2(Omega)$. Furthermore, when $omega_0 in L^infty(Omega)$, the whole sequence is shown to be strongly convergent. This is the first convergence result in numerical approximations of this general class of discontinuous flows. Some important flows such as vortex patches belong to this class.

In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs (digraphs). We obtain a new notion of the path homology and cohomology of a digraph.

We consider a one-dimensional model biodegradation system consisting of two reactionadvection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (<i>pdf</i>) and the traveling front profiles. The travel time <i>pdf</i> does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles