Global well-posedness of 3D two-fluid type model with vacuum: smallness on scaling invariant quantity

Abstract

This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility constraint, and that the newly introduced scaling-invariant initial quantities ${\bar{P}}^{\frac{3}{\gamma }}\left(\Vert \sqrt{{\rho }_0}{u}_0{\Vert }_{L^2}^2+\Vert P_0{\Vert }_{L^1}\right)\left(\Vert \nabla u_0{\Vert }_{L^2}^2+\Vert P_0{\Vert }_{L^2}^2\right)$ and ${\bar{P}}^{\frac{6}{\gamma }+1}{\left(\Vert \sqrt{{\rho }_0}{u}_0{\Vert }_{L^2}^2+\Vert P_0{\Vert }_{L^1}\right)}^3\left(\Vert \nabla u_0{\Vert }_{L^2}^2+\Vert P_0{\Vert }_{L^2}^2\right)$ are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.