Well-posedness of the transport equation by stochastic perturbation

• 成果类型: Article
• 署名作者: Flandoli, F.; Gubinelli, M.; Priola, E.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Universite Paris-Dauphine; University of Pisa; University of Turin
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0224-4
• 发表日期: 2010
• 页码: 1-53
• 关键词: wong-zakai approximations; differential-equations; uniqueness; EXISTENCE; FLOWS; sdes; bv

摘要

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.