Time-periodic solution to a three-phase model of viscoelastic fluid flow

Abstract

<p style='text-indent:20px;'>This paper is concerned with a three-phase model of viscoelastic fluid flow in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ N\geq 5 $\end{document}</tex-math></inline-formula>. We first prove the existence of the time-periodic solution to the integral system in the space of <inline-formula><tex-math id="M4">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>. Then we further show the existence, uniqueness and regularity of the mild solution of the problem. Finally we confirm that such a mild solution is a strong solution in the space of <inline-formula><tex-math id="M5">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>. The proof is based on the compactness analysis with some new development, where a new estimate scheme is artfully constructed.</p>