[Paper Review] Existence and regularity of solution for a Stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion

Dimitra C. Antonopoulou, Geogia Karali+1 more|arXiv (Cornell University)|Oct 4, 2013

Advanced Mathematical Modeling in Engineering11 references3 citations

TL;DR

This paper establishes existence and path regularity for the stochastic Cahn-Hilliard/Allen-Cahn equation with multiplicative space-time white noise and unbounded diffusion coefficient σ satisfying sub-linear growth (|σ(u)| ≤ C(1 + |u|^α) for α ∈ (0, 1/9)). Using semigroup theory and factorization methods, it proves that solution regularity depends on the initial condition: trajectories are almost surely H"older continuous in space with exponent β ∧ (2 − d)/2 and in time with exponent β/4 ∧ (1/2 − d/8), extending prior results to unbounded σ.

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 Petrovski, the bi-Laplacian operator seems to be dominant in the combined model.

Motivation & Objective

• To establish existence and path regularity of solutions for a stochastic Cahn-Hilliard/Allen-Cahn equation with multiplicative space-time white noise.
• To extend previous results—previously valid only for bounded diffusion coefficients—to the case of unbounded diffusion coefficients with sub-linear growth.
• To characterize the regularity of solution trajectories in terms of the regularity of the initial condition and the spatial dimension d.
• To demonstrate that the bi-Laplacian operator dominates the regularity properties in the combined model, consistent with Petrovskii's theory for parabolic operators.

Proposed method

• Employing semigroup theory to analyze the stochastic evolution equation in a Hilbert space framework.
• Using Galerkin approximations to construct approximate solutions and establish convergence.
• Applying the factorization method to handle the stochastic integral with unbounded σ, enabling improved regularity estimates.
• Utilizing Burkholder's inequality and Garsia-Rodemich-Rumsey lemma to derive moment estimates and H"older continuity of the solution paths.
• Introducing stopping times Tn to localize the solution and control the growth of σ(un) via moment bounds under Condition (C) and (\tilde{C}_α).
• Proving that the stochastic integral L(x,t) belongs almost surely to the H"older space Cλ,μ([0,T]×D) for λ < 1/2 − d/8 and μ < 2 − d/2.

Experimental results

Research questions

• RQ1 Can existence and path regularity be established for the stochastic Cahn-Hilliard/Allen-Cahn equation when the noise diffusion coefficient σ is unbounded but satisfies sub-linear growth?
• RQ2 How does the regularity of the solution trajectories depend on the regularity of the initial condition u0 and the spatial dimension d?
• RQ3 What is the optimal H"older regularity in space and time for the solution when σ is Lipschitz and sub-linearly growing?
• RQ4 Does the bi-Laplacian operator dominate the regularity properties in the combined Cahn-Hilliard/Allen-Cahn model, as expected from parabolic theory?
• RQ5 Can the factorization method improve path regularity estimates compared to prior approaches in the bounded diffusion case?

Key findings

• The solution to the stochastic Cahn-Hilliard/Allen-Cahn equation exists globally and is pathwise continuous under the given assumptions on σ and u0.
• If the initial condition u0 is continuous, then the solution has almost surely continuous trajectories in space and time.
• If u0 is β-H"older continuous for β ∈ (0,1), then the solution trajectories are almost surely H"older continuous in space with exponent β ∧ (2 − d)/2 and in time with exponent β/4 ∧ (1/2 − d/8).
• The path regularity of the solution is strictly less than 2 − d/2 in space and 1/2 − d/8 in time when the initial condition vanishes.
• The bi-Laplacian operator dominates the regularity properties of the solution, consistent with the theory of parabolic operators in the sense of Petrovskii.
• The results extend previous work on bounded diffusion coefficients to the more general case of unbounded σ with sub-linear growth, using refined stochastic analysis techniques.