Existence and regularity of solution for a stochastic Cahn–Hilliard/Allen–Cahn equation with unbounded noise diffusion

Abstract

Abstract. The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diffusion coefficient of sub-linear growth. Using technics from semigroup theory, we prove existence, and path regularity of stochastic solution depending on that of the initial condition. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diffusion, for which previous results were established only in the framework of a bounded diffusion coefficient. We prove that the path regularity of stochastic solution depends on that of the initial condition, and are identical to those proved for the stochastic Cahn-Hilliard equation and a bounded noise diffusion coefficient. If the initial condition vanishes, they are strictly less than 2 − d 2 in space and 1 2 − d 8 in time. As expected from the theory of parabolic operators in the sense of Petrovskȋı, the bi-Laplacian operator seems to be dominant in the combined model.

The Stochastic Viscous Cahn–Hilliard Equation: Well-Posedness, Regularity and Vanishing Viscosity Limit

“…Let (u εδ , w εδ , ξ εδ ) and (u δ , w δ , ξ δ ) be any strong solutions with respect to the data (u 0ε , g ε , B εδ ) and (u 0 , g, B δ ), in the cases ε > 0 and ε = 0, respectively: since the first solution component is unique, note that u δ and u εδ are uniquely determined. Since we have already proved the convergence result under the stronger assumption (3.1), we have that u εδ → u δ in L p (Ω; L 2 (0, T ; V 1 )) for every p ∈ [1,2) and every δ > 0, as ε ց 0. Recalling the compatibility condition (2.11) and the fact that (I − δ∆) −2 preserves the mean, by the continuous dependence property of Theorem 2.2 we have H)), the second term on the right-hand side can be made arbitrarily small choosing δ small enough.…”

“…Then the problem (1.1)-(1.4) admits a strong solution. Furthermore, for every p ∈ [1,2] there exists a constant K > 0, independent of ε, such that if (u 1 0 , g 1 , B 1 ) and (u 2 0 , g 2 , B 2 ) satisfy (H1)-(H4), (2.1)-(2.2) and…”

“…Hence, by the continuous dependence result with additive noise case and the Lipschitz continuity of B we have, for every T 0 ∈ (0, T ] and p ∈ [1,2],…”

“…where the implicit constant is independent of ε. In order to prove the result in general for p ∈ [1,2] it is enough to take the p/2-power in Itô's formula, and proceed in the same way, getting…”

“…Existence, uniqueness and regularity have been investigated in the works [22,24,27] for the stochastic pure Cahn-Hilliard equation with a polynomial double-well potential, and in [25,39,62] for the pure case with a possibly singular double-well potential. The reader can also refer to [1] for a study of a stochastic Cahn-Hilliard equation with unbounded noise and [25,26,39] dealing with stochastic Cahn-Hilliard equations with reflections. To the best of our knowledge, the only available results dealing with the stochastic viscous Cahn-Hilliard equation seem to be [41,45]: here, the authors prove existence of mild solution and attractors under the classical case of a polynomial double-well potential.…”