Well-posedness of the transport equation by stochastic perturbation
It is known at the moment that the linear transport equation in hydrodynamics may have multiple solutions of a Cauchy problem in some functional spaces. The main ideological result of the paper under consideration is that under a suitable random perturbation the \(L^\infty\) solutions are unique even in the case where the Cauchy problem for the unperturbed deterministic equation has multiple solutions. This result is obtained by introducing a special multiplicative white noise into the PDE. The assumption on the PDE is that the vector filed is globally Hölder continuous and bounded and satisfies the integrability condition for its divergence.
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