Strong solutions for the stochastic Cahn-Hilliard-Navier-Stokes system. (English) Zbl 1455.35306
Summary: A well-known diffuse interface model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a convective Cahn-Hilliard type equation for the order (phase) parameter. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a stochastic version of this model forced by a multiplicative white noise on a bounded domain of \(\mathbb{R}^d, d = 2, 3\). We prove the existence and uniqueness of a local maximal strong solution when the initial data \(( u_0, \phi_0)\) takes values in \(H^1 \times H^2\). Moreover in the two-dimensional case, we prove that our solution is global.
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35Q30 | Navier-Stokes equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
incompressible two-phase flow; strong pathwise solution; incompressible isothermal mixture of binary fluidsReferences:
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