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Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

Jintao Wang, Chunqiu Li, Lu Yang, Mo Jia

Abstract

In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space $H_0^2(U)$, as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.

Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

Abstract

In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space , as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.

Paper Structure

This paper contains 20 sections, 26 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Existence and Uniqueness of Solutions
  3. Basic notations and properties
  4. Existence of global solutions
  5. Upper Semi-continuity of Random Attractors
  6. Preliminaries on random attractors
  7. Estimates for the solutions
  8. Estimates on solutions for the case of positive kernel
  9. Estimates on solutions for the case of non-negative kernel
  10. Upper semi-continuity of random attractors
  11. The case of positive kernel
  12. The case of non-negative kernel
  13. Existence of Ergodic Invariant Measures
  14. Preliminaries for ergodic invariant measures
  15. The case of positive kernel
  16. ...and 5 more sections

Key Result

Proposition \oldthetheorem

Let ${\mathcal{D}}$ be an inclusion-closed collection of random subsets of $X$ and $\Phi$ a continuous RDS on $X$ over $(\Omega,{\mathcal{F}},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$. Suppose that $\{K(\omega)\}_{\omega\in\Omega}$ is a closed random absorbing set for $\Phi$ in ${\mathcal{D}}$ and

Theorems & Definitions (49)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 39 more