{"title":"The transition to instability for stable shear flows in inviscid fluids","authors":"Daniel Sinambela, 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><</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><</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>></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 with . More precisely, we study the Rayleigh operator associated with perturbed shear flow in a finite channel where with being a stable monotonic shear flow and being a family of perturbations parameterized by m. We prove that there exists such that for , the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when . Moreover, at the nonlinear level, we show that asymptotic instability holds for m near and growing modes exist for which equivalently leads to instability.
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