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Random attractors for damped stochastic fractional Schrödinger equation on $\mathbb{R}^{n}$

Li Lin, Yanjie Zhang, Ao Zhang

TL;DR

The paper analyzes the damped stochastic fractional nonlinear Schrödinger equation on $\mathbb{R}^n$ with fractional Laplacian $(-\Delta)^{\alpha}$ and multiplicative Gaussian noise, focusing on global well-posedness in $H^{\alpha}(\mathbb{R}^n)$ and the emergence of random attractors. Using radial local theory, stochastic Strichartz estimates, and energy/mass methods, it proves global existence and uniqueness under the constraint $\sigma n=2\alpha$ and Assumption $(\mathbf{A_f})$, and constructs a $\mathcal{D}$-pullback random attractor via uniform tail estimates and asymptotic compactness. The work establishes a random dynamical systems framework for the equation on unbounded domains, providing rigorous long-time behavior results for a damped, stochastic, fractional dispersive model. These contributions advance understanding of stochastic dispersive equations with fractional operators and broaden tools for analyzing random attractors in infinite-dimensional settings.

Abstract

We study the random attractors associated with the stochastic fractional Schrödinger equation on $\mathbb{R}^n$. Utilizing the stochastic Strichartz estimates for the damped fractional Schrödinger equation with Gaussian noise, we show the existence and uniqueness of a global solution to the damped stochastic fractional nonlinear Schrödinger equation in $H^α(\mathbb{R}^n)$. Furthermore, we demonstrate that this equation defines an infinite-dimensional dynamical system, which possesses a global attractor in $H^α(\mathbb{R}^n)$.

Random attractors for damped stochastic fractional Schrödinger equation on $\mathbb{R}^{n}$

TL;DR

The paper analyzes the damped stochastic fractional nonlinear Schrödinger equation on with fractional Laplacian and multiplicative Gaussian noise, focusing on global well-posedness in and the emergence of random attractors. Using radial local theory, stochastic Strichartz estimates, and energy/mass methods, it proves global existence and uniqueness under the constraint and Assumption , and constructs a -pullback random attractor via uniform tail estimates and asymptotic compactness. The work establishes a random dynamical systems framework for the equation on unbounded domains, providing rigorous long-time behavior results for a damped, stochastic, fractional dispersive model. These contributions advance understanding of stochastic dispersive equations with fractional operators and broaden tools for analyzing random attractors in infinite-dimensional settings.

Abstract

We study the random attractors associated with the stochastic fractional Schrödinger equation on . Utilizing the stochastic Strichartz estimates for the damped fractional Schrödinger equation with Gaussian noise, we show the existence and uniqueness of a global solution to the damped stochastic fractional nonlinear Schrödinger equation in . Furthermore, we demonstrate that this equation defines an infinite-dimensional dynamical system, which possesses a global attractor in .

Paper Structure

This paper contains 12 sections, 13 theorems, 115 equations.

Table of Contents

  1. Introduction.
  2. Statement of the main results
  3. Assumptions and preliminaries
  4. Main results
  5. Existence of global solution
  6. Radial local theory
  7. Proof of Theorem \ref{['theorem1']}
  8. Existence of random attractors in $H^{\alpha}(\mathbb{R}^n)$
  9. Uniform tail estimates
  10. Existence of random attractor
  11. Proof of Theorem \ref{['theorem2']}
  12. Acknowledgments

Key Result

Theorem 1

Let $n \geq 2$, $\alpha$ be in the interval $\left(\frac{n}{2 n-1}, 1\right)$ and $0<\sigma<\frac{2 \alpha}{n-2 \alpha}$. Let Then under Assumptions ($\bf{A_{f}}$) and $\sigma n=2\alpha$, for any $u_0 \in H^{\alpha}$ radial, there exists a unique global solution of equation 0lin in $H^{\alpha}$, i.e., $\tau^{\ast}(u_0)=\infty$.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 16 more