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

Abstract

This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions $$\begin{array}{c}\left\{\begin{array}{ll}{\mathit{u}}_{\mathit{t}}+\mathit{u}·\nabla \mathit{u}+\nabla \mathit{p}+\mathit{\nu }{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{\beta }}\mathit{u}=\mathit{\theta }\mathit{f}\mathrm{,}& \mathit{x}\in {\mathbb{R}}^{\mathit{n}}\mathrm{,}\mathit{t}\mathrm{>}0\mathrm{,}\\ {\mathit{\theta }}_{\mathit{t}}+\mathit{u}·\nabla \mathit{\theta }+{\mathit{g}}_{\mathit{\alpha }}\ast {\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{\gamma }}\mathit{\theta }=0\mathrm{,}& \mathit{x}\in {\mathbb{R}}^{\mathit{n}}\mathrm{,}\mathit{t}\mathrm{>}0\mathrm{,}\\ div\mathit{u}=0\mathrm{,}& \mathit{x}\in {\mathbb{R}}^{\mathit{n}}\mathrm{,}\mathit{t}\mathrm{>}0\mathrm{,}\\ \mathit{u}\mathrm{(}\mathit{x}\mathrm{,}0\mathrm{)}={\mathit{u}}_0\mathrm{,}\mathit{\theta }\mathrm{(}\mathit{x}\mathrm{,}0\mathrm{)}={\mathit{\theta }}_0\mathrm{,}& \mathit{x}\in {\mathbb{R}}^{\mathit{n}}\mathrm{,}\end{array}\right.\end{array}$$ where ${\mathit{g}}_{\mathit{\alpha }}\mathrm{(}\mathit{t}\mathrm{)}=\frac{{\mathit{t}}^{\mathit{\alpha }-1}}{\mathrm{\Gamma }\mathrm{(}\mathit{\alpha }\mathrm{)}}$ . 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 $\mathit{\gamma }=\mathrm{(}\mathit{\alpha }+1\mathrm{)}\mathit{\beta }$ . Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.