Existence and smoothness of the density for the stochastic continuity
  equation

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Abstract

We consider the stochastic continuity equation driven by Brownian motion. We use the techniques of the Malliavin calculus to show that the law of the solution has a density with respect to the Lebesgue measure. We also prove that the density is Holder continuous and satisfies some Gaussian-type estimates.

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Hyperbolic Conservation Laws in Continuum Physics

Constantine Dafermos (2010)

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Noise Prevents Singularities in Linear Transport Equations

Ennio Fedrizzi, Franco Flandoli (2012)

A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial conditions may develop discontinuities, we prove that a certain Sobolev degree of regularity is maintained, which implies H\"older continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.