Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition

  • Wladimir Neves
  • Christian OliveraEmail author


We consider the stochastic divergence-free continuity equations with Ladyzhenskaya–Prodi–Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of strong uniqueness, in the probabilistic sense, relies on stochastic characteristic method and the generalized Itô–Wentzell–Kunita formula. The stability property for the unique solution is proved with respect to the initial data. Moreover, a persistence result is established by a representation formula.


Stochastic partial differential equation Continuity equation Well-posedness Stochastic characteristic method Cauchy problem 

Mathematics Subject Classification

60H15 35R60 35F10 60H30 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Departamento de MatemáticaUniversidade Estadual de CampinasCampinasBrazil

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