Inventiones mathematicae

, Volume 180, Issue 1, pp 1–53 | Cite as

Well-posedness of the transport equation by stochastic perturbation

Article

Abstract

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica Applicata “U. Dini”Università di PisaPisaItaly
  2. 2.CEREMADE (UMR 7534)Université Paris DauphineParis cedex 16France
  3. 3.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

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