Noise Prevents Singularities in Linear Transport Equations

Ennio Fedrizzi (1) , Franco Flandoli (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Dipartimento di Matematica

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.

Mots clés: stochastic linear trasport equation; no-blow-up; integrable drift; Sobolev initial conditions

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Ennio Fedrizzi, Franco Flandoli. Noise Prevents Singularities in Linear Transport Equations. Journal of Functional Analysis, 2013, 264 (6), pp.1329-1354. ⟨10.1016/j.jfa.2013.01.003⟩. ⟨hal-01017915⟩