On the Boussinesq system with fractional memory in pseudo-measure spaces

Article type: Research Article

Authors: Poblete, Felipe^a | Silva, Clessius^b | Viana, Arlúcio^{c; *}

Affiliations: [a] Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Chile | [b] Departamento de Matemática, Universidade Federal Rural de Pernambuco, Brazil | [c] Departamento de Matemática, Universidade Federal de Sergipe, Brazil

Correspondence: [*] Corresponding author. E-mail: arlucioviana@academico.ufs.br.

Abstract: This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions ut+u·∇u+∇p+ν(−Δ)βu=θf,x∈Rn,t>0,θt+u·∇θ+gα∗(−Δ)γθ=0,x∈Rn,t>0,divu=0,x∈Rn,t>0,u(x,0)=u0,θ(x,0)=θ0,x∈Rn, where gα(t)=tα−1Γ(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.

Keywords: Nonlocal Navier–Stokes, Boussinesq system, PDEs in connection with fluid mechanics, fractional memory, self-similarity

DOI: 10.3233/ASY-241904

Journal: Asymptotic Analysis, vol. 139, no. 3-4, pp. 157-181, 2024