{"title":"Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: Well-posedness and maximal regularity","authors":"Pascal Auscher , Pierre Portal","doi":"10.1016/j.jde.2024.11.038","DOIUrl":null,"url":null,"abstract":"<div><div>We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>, for all <span><math><mn>1</mn><mo>−</mo><mi>ε</mi><mo><</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span> in this generality.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"420 ","pages":"Pages 1-51"},"PeriodicalIF":2.4000,"publicationDate":"2024-12-02","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/S0022039624007678","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 well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces . Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on to , for all in this generality.
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