A representation of solutions to a scalar conservation law in several dimensions

Details A representation of solutions to a scalar conservation law in several dimensions A representation of solutions to a scalar conservation law in several dimensions

2012-03-17

arxiv.org/abs/1203.3901v1

Mathematics

Analysis of PDEs

11 pages

Scientific paper

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of singularities formation. Further, we introduce an associated system of balance laws that can be interpreted as describing the motion of a continuum with some specific pressure term. This term arises only after the instant when the solution to the initial Cauchy problem looses its smoothness. Before this instant the system coincides partly with the one known as pressure free gas dynamics.

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