Well-posedness for the Cahn-Hilliard-Navier-Stokes equations perturbed by gradient-type noise, in two dimensions. (English) Zbl 1537.35436
Summary: In this work, we study the problem of existence and uniqueness of solutions of the stochastic Cahn-Hilliard-Navier-Stokes system with gradient-type noise. We show that such kind of noise is related to the problem of modelling turbulence. We apply a rescaling argument to transform the stochastic system into a random deterministic one. We split the latter into two parts: the Navier-Stokes part and the Cahn-Hilliard part, respectively. The rescale operators possess good properties which allow to show that the rescaled Navier-Stokes equations have a unique solution, by appealing to \(\delta\)-monotone operators theory. While, well-posedness of the Cahn-Hilliard part is proved via a fixed point argument. Then, again a fixed point argument is used to prove global in time existence of a unique solution to the initial system. All the results are under the requirement that the initial data is in a certain small neighbourhood of the origin.
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D06 | Statistical solutions of Navier-Stokes and related equations |
| 80A22 | Stefan problems, phase changes, etc. |
Keywords:
Cahn-Hilliard-Navier-Stokes equations; existence and uniqueness; gradient-type noise; M-accretive operator; fixed point argumentReferences:
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