Noise Prevents Singularities in Linear Transport Equations

Ennio Fedrizzi; Franco Flandoli

doi:10.1016/j.jfa.2013.01.003

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arXiv · 1205.5505

Noise Prevents Singularities in Linear Transport Equations

Ennio Fedrizzi·Franco Flandoli

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Abstract

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 Hölder continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.

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Ennio Fedrizzi, Franco Flandoli. 2012-05-24. Noise Prevents Singularities in Linear Transport Equations. https://doi.org/10.1016/j.jfa.2013.01.003

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