Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Congcong Li; Chunqiu Li; Jintao Wang

doi:10.3934/dcdsb.2021311

Article Contents

2022, Volume 27, Issue 10: 6173-6196. Doi: 10.3934/dcdsb.2021311

This issue Previous Article Multi-valued random dynamics of stochastic wave equations with infinite delays Next Article Boundary layer effects on ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

1.
Department of Data science and Big data technology, Wenzhou University, Wenzhou, Zhejiang 325035, China
2.
Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang 325035, China

^* Corresponding author: Chunqiu Li
^* Corresponding author: Chunqiu Li

Received: August 2021

Revised: November 2021

Early access: January 2022

Published: October 2022

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Abstract

In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-$ {\mathcal{D}} $ attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-$ {\mathcal{D}} $ attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.

Keywords:

Statistical solution,
invariant measure,
pullback attractor,
discrete coupled Schrödinger-Boussinesq equations.

Mathematics Subject Classification: 35B41, 35D99, 76F20.

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