Well-posedness of the transport equation by stochastic perturbation

作者：Flandoli F; Gubinelli M^*; Priola E

来源：Inventiones Mathematicae, 2010, 180(1): 1-53.

DOI：10.1007/s00222-009-0224-4

We consider the linear transport equation with a globally Holder 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-Tanaka type.

• 出版日期2010-4