{"title":"On well/ill-posedness for the generalized surface quasi-geostrophic equation in Hölder spaces","authors":"Young-Pil Choi , Jinwook Jung , Junha Kim","doi":"10.1016/j.jde.2025.113521","DOIUrl":null,"url":null,"abstract":"<div><div>We establish the well/ill-posedness theories for the inviscid <em>α</em>-surface quasi-geostrophic (<em>α</em>-SQG) equation in Hölder spaces, where <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>=</mo><mn>1</mn></math></span> correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of <em>α</em>-SQG equation in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>β</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span> with <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> for some <span><math><mi>T</mi><mo>></mo><mn>0</mn></math></span>. We then analyze the strong ill-posedness in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>α</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> constructing smooth solutions to the <em>α</em>-SQG equation that exhibit <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>–norm growth in a short time. In particular, we develop the nonexistence theory for <em>α</em>-SQG equation in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>α</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113521"},"PeriodicalIF":2.3000,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625005480","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
We establish the well/ill-posedness theories for the inviscid α-surface quasi-geostrophic (α-SQG) equation in Hölder spaces, where and correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of α-SQG equation in with for some . We then analyze the strong ill-posedness in constructing smooth solutions to the α-SQG equation that exhibit –norm growth in a short time. In particular, we develop the nonexistence theory for α-SQG equation in .
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics