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 $\bar P^{\frac{ 3}γ} \left(|\sqrt{ρ0}u_0|{L^2}^2+|P_0|_{L^1}\right) \left(|\nabla u_0|{L^2}^2+|P_0|{L^2}^2\right)$ and $\bar P^{\frac{6}γ+1} \left(|\sqrt{ρ0}u_0|{L^2}^2+|P_0|_{L^1}\right)^3 \left(|\nabla u_0|{L^2}^2+|P_0|{L^2}^2\right)$ are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.