The transition to instability for stable shear flows in inviscid fluids

IF 1.7 2区 数学 Q1 MATHEMATICS
Daniel Sinambela, Weiren Zhao
{"title":"The transition to instability for stable shear flows in inviscid fluids","authors":"Daniel Sinambela,&nbsp;Weiren Zhao","doi":"10.1016/j.jfa.2025.110905","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> with <span><math><mi>s</mi><mo>&lt;</mo><mn>2</mn></math></span>. More precisely, we study the Rayleigh operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>γ</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>γ</mi></mrow></msub><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>−</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>γ</mi></mrow><mrow><mo>″</mo></mrow></msubsup><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><msup><mrow><mi>Δ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> associated with perturbed shear flow <span><math><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>γ</mi></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> in a finite channel <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn><mi>π</mi></mrow></msub><mo>×</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> where <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>γ</mi></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>U</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>m</mi><msup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>Γ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>y</mi><mo>/</mo><mi>γ</mi><mo>)</mo></math></span> with <span><math><mi>U</mi><mo>(</mo><mi>y</mi><mo>)</mo></math></span> being a stable monotonic shear flow and <span><math><msub><mrow><mo>{</mo><mi>m</mi><msup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>Γ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>y</mi><mo>/</mo><mi>γ</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> being a family of perturbations parameterized by <em>m</em>. We prove that there exists <span><math><msub><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> such that for <span><math><mn>0</mn><mo>≤</mo><mi>m</mi><mo>&lt;</mo><msub><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when <span><math><mi>m</mi><mo>≥</mo><msub><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>. Moreover, at the nonlinear level, we show that asymptotic instability holds for <em>m</em> near <span><math><msub><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> and growing modes exist for <span><math><mi>m</mi><mo>&gt;</mo><msub><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> which equivalently leads to instability.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 2","pages":"Article 110905"},"PeriodicalIF":1.7000,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123625000874","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in Cs with s<2. More precisely, we study the Rayleigh operator LUm,γ=Um,γxUm,γxΔ1 associated with perturbed shear flow (Um,γ(y),0) in a finite channel T2π×[1,1] where Um,γ(y)=U(y)+mγ2Γ˜(y/γ) with U(y) being a stable monotonic shear flow and {mγ2Γ˜(y/γ)}m0 being a family of perturbations parameterized by m. We prove that there exists m such that for 0m<m, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when mm. Moreover, at the nonlinear level, we show that asymptotic instability holds for m near m and growing modes exist for m>m which equivalently leads to instability.
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
3.20
自引率
5.90%
发文量
271
审稿时长
7.5 months
期刊介绍: The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信