Well-posedness of the transport equation by stochastic perturbation

@article{Flandoli2008WellposednessOT,
  title={Well-posedness of the transport equation by stochastic perturbation},
  author={Franco Flandoli and Massimiliano Gubinelli and Enrico Priola},
  journal={Inventiones mathematicae},
  year={2008},
  volume={180},
  pages={1-53}
}
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… 
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