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

TL;DR

This paper proves the global existence and uniqueness of solutions for a 3D two-fluid model with vacuum, under smallness conditions on certain scaling-invariant initial quantities.

Contribution

It establishes the first global well-posedness result for the 3D two-fluid model allowing vacuum with small initial data in scaling-invariant norms.

Findings

Global well-posedness of strong solutions is achieved.

Smallness in scaling-invariant quantities ensures solution existence.

The results extend understanding of two-fluid models with vacuum.

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 $\bar P^{\frac{ 3}{\gamma}} \left(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|P_0\|_{L^1}\right) \left(\|\nabla u_0\|_{L^2}^2+\|P_0\|_{L^2}^2\right)$ and $\bar P^{\frac{6}{\gamma}+1} \left(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|P_0\|_{L^1}\right)^3 \left(\|\nabla u_0\|_{L^2}^2+\|P_0\|_{L^2}^2\right)$ are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.