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

Deugoué, G.; Ngana, A. Ndongmo; Medjo, T. Tachim

doi:10.1007/s00021-020-00503-9

Cited by 5 publications

(3 citation statements)

References 35 publications

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“…To present ideas clearly, we simply do estimations on g(u) instead of g(t, u). Indeed, the idea and procedures to obtain existence of martingale solutions to (1.1) with g(t, u) are similar, we only need to consider for every t ∈ (τ, T ], ũ(t) is Ft -measurable, for more details, see [13].…”

Section: Stochastic Settingmentioning

confidence: 99%

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Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

Xu¹,

Caraballo²

2022

SIAM J. Math. Anal.

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This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo-Galerkin approximation and an idea analogous to Da Prato and Zabczyk [12]. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solutions of the deterministic problem is also solution of the stochastic one, or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong-Zakai's technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong-Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.

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Section: Stochastic Settingmentioning

confidence: 99%

“…This theory has received increasing attention over the last years (see, e.g. [12,13,14,26]). In what follows, we introduce some necessary notations and most of the hypotheses relevant to our analysis.…”

mentioning

confidence: 99%

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

Xu¹,

Caraballo²

2022

SIAM J. Math. Anal.

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“…Besides, let us point out the mathematical literature on stochastic two-phase flows has also been expanded in the last years, in the context of coupled stochastic systems of Cahn-Hilliard-Navier-Stokes and Allen-Cahn-Navier-Stokes type. In this direction, we refer the reader to the contributions [33,35,80,81] for existence of solutions, [3,36] about asymptotic long-time behaviour, [37] on large deviation limits, and [32,34] dealing with a nonlocal phase-field equation in the system instead.…”

Section: Introductionmentioning

confidence: 99%

The stochastic Cahn–Hilliard equation with degenerate mobility and logarithmic potential

Scarpa

2021

Nonlinearity

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We prove existence of martingale solutions for the stochastic Cahn–Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by Elliott and Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed.

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Solution to a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes model with shear dependent viscosity via a splitting-up method

Deugoué

Moghomye

Medjo

2022

Nonlinear Differ. Equ. Appl.

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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

Cited by 5 publications

References 35 publications

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

The stochastic Cahn–Hilliard equation with degenerate mobility and logarithmic potential

Solution to a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes model with shear dependent viscosity via a splitting-up method

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