On the weak solutions to a stochastic two-phase flow model. (English) Zbl 1485.35442
Summary: We study, in this article, a stochastic version of a coupled Allen-Cahn-Navier-Stokes model in a two- or three-dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. We prove the existence of a probabilistic weak solutions. The proof relies on a Galerkin approximation as well as some compactness results. In the two-dimensional case, we prove the uniqueness of the weak solutions.
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35D30 | Weak solutions to PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
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