Local Existence and Uniqueness of Strong Solution to the Inhomogeneous Primitive Equations with Vacuum.

In this paper, we establish the existence and uniqueness of the local strong solution to the Cauchy problem of the inhomogeneous incompressible primitive equations in a periodic domain. Inspired by the work (J. Li in J. Differ. Equ. 263:6512–6536, 2017), we remove the compatibility condition required in (Q. Jiu and F. Wang in Acta Math. Sci. Ser. B Engl. Ed. 40:1316–1334, 2020), and permit the initial density with vacuum. Meanwhile, we prove the existence of the eigenfunctions to the hydrostatic Stokes operator. [ABSTRACT FROM AUTHOR]