{"title":"Ill/well-posedness of non-diffusive active scalar equations with physical applications","authors":"","doi":"10.1016/j.jde.2024.08.062","DOIUrl":null,"url":null,"abstract":"<div><p>We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator <strong>T</strong> that is singular of order <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. For <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> we prove well-posedness in Gevrey spaces <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> with <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac><mo>)</mo></math></span>, while for <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> and further conditions on <strong>T</strong> we prove ill-posedness in <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> for suitable <em>s</em>. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-09-06","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/S0022039624005527","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 consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator T that is singular of order . For we prove well-posedness in Gevrey spaces with , while for and further conditions on T we prove ill-posedness in for suitable s. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.
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