Stability of strong solutions for a model of incompressible two–phase flow under thermal fluctuations

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

“…Moreover, assumption ND2 allows of course for the constant mobility scenario, but also includes the case of positive nonconstant mobilities. Condition ND3 on the noise is widely employed in literature (see for example [7,35,36]), and ensures that in particular that G : H → L 2 (U, H) is Lipschitz-continuous and linearly bounded, and that the restriction…”

“…The mathematical literature on stochastic phase-field modelling has also been increasingly developed. Let us point out in this direction the works [1] dealing with unbounded noise, [35,36] for a study of a diffuse interface model with thermal fluctuations, and [4,60] dealing with the stochastic Allen-Cahn equation. Beside well-posedness, optimal control problems have also been studied in [66] in the case of the stochastic Cahn-Hilliard equation, and in [59] in the context of a stochastic phase-field model for tumour growth.…”

Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

“…Some of the results in [6] and the current paper have already been used in several papers such as [9], [10], [65], [30], [29] and [64]. Very recently we have become aware of a recent paper by Feireisl and Petcu [26], in which they proved the existence of a dissipative martingale, as well as the existence of a local strong solution and weak-strong uniqueness of the solution of the stochastic Navier-Stokes Allen-Cahn Equations. Note that in [26] the second unknown n is a scalar field, the nonlinear term f (·) is globally Lipschitz and the derivative of a double-well potential F (·), and the coefficient of the noise entering the equations for n is bounded.…”

“…Very recently we have become aware of a recent paper by Feireisl and Petcu [26], in which they proved the existence of a dissipative martingale, as well as the existence of a local strong solution and weak-strong uniqueness of the solution of the stochastic Navier-Stokes Allen-Cahn Equations. Note that in [26] the second unknown n is a scalar field, the nonlinear term f (·) is globally Lipschitz and the derivative of a double-well potential F (·), and the coefficient of the noise entering the equations for n is bounded. The paper [5] is the first paper to deal with the the stochastic counterpart of the Ericksen-Leslie equations (1.1)- (1.4).…”

Optimal control of stochastic phase-field models related to tumor growth

“…The mathematical literature on the stochastic Cahn-Hilliard and Allen-Cahn equations is quite developed: we refer for example to [3,47] for the stochastic Allen-Cahn equation, and to [12,15,16,20,31,50,51]) for the stochastic Cahn-Hilliard equation. For completeness, let us quote also [21,22] for a study on a stochastic diffuse interface model involving the Cahn-Hilliard and Navier-Stokes equations. Nevertheless, similar results for coupled stochastic Cahn-Hilliard reaction-diffusion systems were not previously studied, up to our knowledge: in this sense, this contribution can be seen as a first work in this direction.…”