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On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model. (English) Zbl 1365.35235

Let \(M\) be a bounded domain in \(\mathbb R^2\) and let \(T>0\) be a constant. Consider the following stochastic Cahn-Hilliard-Navier-Stokes equation on \(M\times [0,T]\): \[ \begin{aligned} &\partial_t v-\theta_1\Delta v +(v\cdot \nabla)v+\nabla p- \kappa \mu \nabla \phi = g_1(t,v)+g_2(t,v)\dot W_t,\\ & \mathrm{div } v=0,\\ &\partial_t \phi +v\cdot \nabla \phi-\theta_3\mu=0,\\ &\mu=-\theta_2\Delta\phi +\alpha f(\phi),\end{aligned} \] where \(\theta_i,\kappa,\alpha\) are positive constants, \(W_t\) is a cylindrical Brwonian motion on a reference Hilbert space, \(g_i, f\) are suitable maps. Under suitable regularity and growth conditions on \(g_i\) and \(f\), the author proves the existence and uniqueness of variational solutions for the unknown velocity \(v=(v_1v_2)\) and order parameter \(\phi\).

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
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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