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Global Existence of Martingale Solutions and Large Time Behavior for a 3D Stochastic Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Shear Dependent Viscosity

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Abstract

In this paper, we consider a stochastic version of a nonlinear system which consists of the incompressible Navier–Stokes equations with shear dependent viscosity controlled by a power \(p>2\), coupled with a convective nonlocal Cahn–Hilliard-equations. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluid having the same density. We prove the existence of a weak martingale solutions when \(p\in [11/5,12/5)\), and their exponential decay when the time goes to infinity.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Dschang, P. O. BOX 67, Dschang, Cameroon

    G. Deugoué & A. Ndongmo Ngana

  2. Department of Mathematics and Statistics, Florida International University, MMC, Miami, FL, 33199, USA

    T. Tachim Medjo

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  1. G. Deugoué
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  2. A. Ndongmo Ngana
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  3. T. Tachim Medjo
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Correspondence to T. Tachim Medjo.

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Deugoué, G., Ngana, A.N. & Medjo, T.T. Global Existence of Martingale Solutions and Large Time Behavior for a 3D Stochastic Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Shear Dependent Viscosity. J. Math. Fluid Mech. 22, 46 (2020). https://doi.org/10.1007/s00021-020-00503-9

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