Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three

The Cauchy problem of the reduced gravity two and a half layer model in dimension three isconsidered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit two kinds of the generalized Huygens' waves. It is a significant different phenomenon from the Navier-Stokes system. Lastly, as a byproduct, we also extend \begin{document}$L^2(\mathbb{R}^3)$\end{document} -decay rate to \begin{document}$L^p(\mathbb{R}^3)$\end{document} -decay rate with \begin{document}$p>1$\end{document} .