{"title":"Exponential Mixing for the White-Forced Complex Ginzburg–Landau Equation in the Whole Space","authors":"Vahagn Nersesyan, Meng Zhao","doi":"10.1137/23m1597150","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3646-3678, June 2024. <br/> Abstract. In the last two decades, there has been significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg–Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently nondegenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş–Prodi estimate to the case of the real line and special space-time weighted estimates, which help to handle the behavior of solutions at infinity.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1597150","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3646-3678, June 2024. Abstract. In the last two decades, there has been significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg–Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently nondegenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş–Prodi estimate to the case of the real line and special space-time weighted estimates, which help to handle the behavior of solutions at infinity.
期刊介绍：
SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena.
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Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.
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