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Martingale Solutions of Stochastic Constrained Modified Swift-Hohenberg Equation

Saeed Ahmed, Javed Hussain

TL;DR

This work establishes the existence of global martingale solutions for the stochastic constrained Modified Swift–Hohenberg equation driven by Stratonovich multiplicative noise in a two-dimensional bounded domain. It adopts a Hilbert-space formulation with a tangent-space projection to enforce the unit-nnorm constraint, then builds finite-dimensional Faedo–Galerkin approximations that preserve the constraint. Uniform energy and moment estimates, tightness via Aldous conditions, and convergence of quadratic variations lead to a Skorokhod-type representation, enabling a martingale representation and construction of a solution. The results provide a rigorous well-posedness framework for stochastic amplitude equations modeling pattern formation under gradient-type noise, with potential implications for stochastic modeling of thin-film and related systems.

Abstract

In this paper, we aim to prove the existence of global Martingale solution to Stochastic Constrained Modified Swift-Hohenberg Equation driven by stratonovich multiplicative noise. This equation belongs to class of amplitude equations which describe the appearance of pattern formation in nature. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzezniak and collaborators, and Jakubowski's generalization of the Skorokhod theorem.

Martingale Solutions of Stochastic Constrained Modified Swift-Hohenberg Equation

TL;DR

This work establishes the existence of global martingale solutions for the stochastic constrained Modified Swift–Hohenberg equation driven by Stratonovich multiplicative noise in a two-dimensional bounded domain. It adopts a Hilbert-space formulation with a tangent-space projection to enforce the unit-nnorm constraint, then builds finite-dimensional Faedo–Galerkin approximations that preserve the constraint. Uniform energy and moment estimates, tightness via Aldous conditions, and convergence of quadratic variations lead to a Skorokhod-type representation, enabling a martingale representation and construction of a solution. The results provide a rigorous well-posedness framework for stochastic amplitude equations modeling pattern formation under gradient-type noise, with potential implications for stochastic modeling of thin-film and related systems.

Abstract

In this paper, we aim to prove the existence of global Martingale solution to Stochastic Constrained Modified Swift-Hohenberg Equation driven by stratonovich multiplicative noise. This equation belongs to class of amplitude equations which describe the appearance of pattern formation in nature. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzezniak and collaborators, and Jakubowski's generalization of the Skorokhod theorem.
Paper Structure (8 sections, 17 theorems, 162 equations)

This paper contains 8 sections, 17 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Stochastic framework and main results
  3. Tangent space and Orthogonal Projection
  4. Martingale Representation Theorem
  5. Faedo Galerkin Approximation
  6. Tightness of Measures
  7. Convergence of Quadratic Variations
  8. Existence of Martingale Solution

Key Result

Theorem 2.3

ref2 The sequence $\{Y_k\}$ of random variables with values in $S$ satisfies [T] if and only if

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2: Modulus of continuity
  • Theorem 2.3
  • Lemma 2.4
  • Definition 2.5: Aldous Condition
  • Proposition 2.6
  • Lemma 2.7
  • Theorem 2.8: Martingale Representation Theorem
  • Theorem 3.1
  • proof
  • ...and 20 more