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Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory

Jiaohui Xu, Tomás Caraballo, José Valero

TL;DR

This work analyzes stochastic nonlocal PDEs with long time memory driven by additive noise, employing a Galerkin scheme combined with Dafermos transformation to obtain well-posedness and a random dynamical framework. It proves the existence of tempered random attractors in the natural phase space and develops Wong-Zakai colored-noise approximations to connect the stochastic model to deterministic limits, establishing convergence of solutions and upper semicontinuity of attractors as the approximation parameter tends to zero. The results build a rigorous link between memory effects, stochastic forcing, and regularized approximations, with an Appendix detailing the deterministic counterpart. These findings provide a solid foundation for analyzing memory-enabled stochastic PDEs and their regularized approximations in applications requiring long-range temporal effects.

Abstract

In this paper, a combination of Galerkin's method and Dafermos' transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.

Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory

TL;DR

This work analyzes stochastic nonlocal PDEs with long time memory driven by additive noise, employing a Galerkin scheme combined with Dafermos transformation to obtain well-posedness and a random dynamical framework. It proves the existence of tempered random attractors in the natural phase space and develops Wong-Zakai colored-noise approximations to connect the stochastic model to deterministic limits, establishing convergence of solutions and upper semicontinuity of attractors as the approximation parameter tends to zero. The results build a rigorous link between memory effects, stochastic forcing, and regularized approximations, with an Appendix detailing the deterministic counterpart. These findings provide a solid foundation for analyzing memory-enabled stochastic PDEs and their regularized approximations in applications requiring long-range temporal effects.

Abstract

In this paper, a combination of Galerkin's method and Dafermos' transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.
Paper Structure (12 sections, 27 theorems, 352 equations)

This paper contains 12 sections, 27 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set-up of the problem
  4. Assumptions
  5. Notation
  6. Well-posedness of problem \ref{['eq1-1']}
  7. An auxiliary result
  8. Well-posedness of problem \ref{['eq1-5']}
  9. Existence of random attractors
  10. Stochastic nonlocal PDEs with long time memory driven by colored noise
  11. Upper-semicontinuity
  12. Appendix

Key Result

Lemma 3.1

Assume $(h_{1})$-$(h_{2})$ hold. Then, the operator $\mathcal{J}:L_{V}^{2}\rightarrow L_{\mu}^{2}(\mathbb{R}^{+};V)$ defined by, is a linear and continuous operator. In particular, there exists a positive constant $K_{\mu}=e^{\gamma}\int_{0}^{1}\mu(s)ds+\mu(1)e^{\varpi}(\gamma-\varpi)^{-2}$ such that for any $\varphi\in L_{V}^{2}$, it holds

Theorems & Definitions (38)

  • Remark 2.1
  • Lemma 3.1
  • Corollary 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 3.6
  • Corollary 3.7
  • Theorem 3.8
  • Remark 3.9
  • ...and 28 more