{"title":"关于伪测量空间中具有分数记忆的布森斯克系统","authors":"Felipe Poblete, Clessius Silva, A. Viana","doi":"10.3233/asy-241904","DOIUrl":null,"url":null,"abstract":"This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ = 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β. Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Boussinesq system with fractional memory in pseudo-measure spaces\",\"authors\":\"Felipe Poblete, Clessius Silva, A. Viana\",\"doi\":\"10.3233/asy-241904\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ = 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β. Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.\",\"PeriodicalId\":55438,\"journal\":{\"name\":\"Asymptotic Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-03-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Asymptotic Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3233/asy-241904\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asymptotic Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3233/asy-241904","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
本文研究了具有分数记忆和分数扩散的布森斯克系统的局部和全局自相似解的存在性 u t + u - ∇ u + ∇ p + ν ( - Δ ) β u = θ f , x ∈ R n 、t > 0 , θ t + u -∇ θ + g α ∗ ( - Δ ) γ θ = 0 , x∈ R n , t > 0 , div u = 0 , x∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x∈ R n , 其中 g α ( t ) = t α - 1 Γ ( α ) 。存在性结果是在伪度量空间框架下得到的。我们发现,全局解的存在性和自相似性深受记忆和分数扩散之间关系的影响。事实上,只有当 γ = ( α + 1 ) β 时,我们才能得到全局解的存在性和自相似性。此外,我们还证明了全局解的稳定性结果和渐近自相似解的存在性。
On the Boussinesq system with fractional memory in pseudo-measure spaces
This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ = 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β. Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.
期刊介绍:
The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand. Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.