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

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 ${\bar{P}}^{\frac{3}{\gamma }}(\mathrm{\parallel }\sqrt{{\rho }_0}{u}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{P}_0{\mathrm{\parallel }}_{L^1})(\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{P}_0{\mathrm{\parallel }}_{L^2}^2)$ and ${\bar{P}}^{\frac{6}{\gamma }+1}{(\mathrm{\parallel }\sqrt{{\rho }_0}{u}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{P}_0{\mathrm{\parallel }}_{L^1})}^3(\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{P}_0{\mathrm{\parallel }}_{L^2}^2)$ are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.