{"title":"Well-posedness for the Cahn-Hilliard-Navier-Stokes Equations Perturbed by Gradient-Type Noise, in Two Dimensions","authors":"Ionuţ Munteanu","doi":"10.1007/s00245-024-10121-w","DOIUrl":null,"url":null,"abstract":"<div><p>In this work, we study the problem of existence and uniqueness of solutions of the stochastic Cahn-Hilliard-Navier-Stokes system with gradient-type noise. We show that such kind of noise is related to the problem of modelling turbulence. We apply a rescaling argument to transform the stochastic system into a random deterministic one. We split the latter into two parts: the Navier-Stokes part and the Cahn-Hilliard part, respectively. The rescale operators possess good properties which allow to show that the rescaled Navier-Stokes equations have a unique solution, by appealing to <span>\\(\\delta -\\)</span>monotone operators theory. While, well-posedness of the Cahn-Hilliard part is proved via a fixed point argument. Then, again a fixed point argument is used to prove global in time existence of a unique solution to the initial system. All the results are under the requirement that the initial data is in a certain small neighbourhood of the origin.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"89 2","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10121-w","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we study the problem of existence and uniqueness of solutions of the stochastic Cahn-Hilliard-Navier-Stokes system with gradient-type noise. We show that such kind of noise is related to the problem of modelling turbulence. We apply a rescaling argument to transform the stochastic system into a random deterministic one. We split the latter into two parts: the Navier-Stokes part and the Cahn-Hilliard part, respectively. The rescale operators possess good properties which allow to show that the rescaled Navier-Stokes equations have a unique solution, by appealing to \(\delta -\)monotone operators theory. While, well-posedness of the Cahn-Hilliard part is proved via a fixed point argument. Then, again a fixed point argument is used to prove global in time existence of a unique solution to the initial system. All the results are under the requirement that the initial data is in a certain small neighbourhood of the origin.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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