Roberta Bianchini, Timothée Crin-Barat, Marius Paicu
{"title":"Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation","authors":"Roberta Bianchini, Timothée Crin-Barat, Marius Paicu","doi":"10.1007/s00205-023-01945-x","DOIUrl":null,"url":null,"abstract":"<div><p>We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in <span>\\(\\dot{H}^{1-\\tau }(\\mathbb {R}^2) \\cap \\dot{H}^s(\\mathbb {R}^2)\\)</span> with <span>\\(s > 3\\)</span> and for any <span>\\(0< \\tau <1\\)</span>. Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to <span>\\(H^{20}(\\mathbb {R}^2)\\)</span>. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in <span>\\(H^{1-\\tau }(\\mathbb {R}^2) \\cap \\dot{H}^s(\\mathbb {R}^2)\\)</span> with <span>\\(s > 3\\)</span> and <span>\\(0< \\tau <1\\)</span>. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity <span>\\(\\Vert u_2(t)\\Vert _{L^\\infty (\\mathbb {R}^2)}\\)</span> for initial data only in <span>\\(\\dot{H}^{1-\\tau }(\\mathbb {R}^2) \\cap \\dot{H}^s(\\mathbb {R}^2)\\)</span> with <span>\\(s >3\\)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-023-01945-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in \(\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)\) with \(s > 3\) and for any \(0< \tau <1\). Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to \(H^{20}(\mathbb {R}^2)\). More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in \(H^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)\) with \(s > 3\) and \(0< \tau <1\). Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity \(\Vert u_2(t)\Vert _{L^\infty (\mathbb {R}^2)}\) for initial data only in \(\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)\) with \(s >3\).
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.