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Well-posedness of the transport equation by stochastic perturbation

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Stochastic Partial Differential Equations (SPDEs)

This is a joint work with F. Flandoli and M. Gubinelli. We consider the linear transport equation with a globally H”{o}lder 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 It^{o}-Tanaka type.

This talk is part of the Isaac Newton Institute Seminar Series series.

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