Low Mach number limit for a compressible two-fluid model with algebraic closure and ill-prepared initial data in critical Besov spaces

RESUMO

In this paper, we study the low Mach number limit for a compressible two-fluid model with algebraic closure in the $d$-dimensional torus with $d \geq 2$. For large and ill-prepared initial data in critical Besov spaces, we prove that, provided that the Mach number is sufficiently small, the rescaled compressible two-fluid flow exists in critical Besov spaces for any finite time not exceeding the lifespan of the incompressible flow. Moreover, the rescaled compressible two-fluid flow converges to the incompressible Navier--Stokes flow as the Mach number tends to zero. The proof is based on a high-middle-low frequency analysis of the densities and velocity field, combined with a filtering technique involving wave operators. The main novelty is the derivation of new a priori estimates for the high-middle frequency part of the solution to the compressible two-fluid model, depending explicitly on time, the frequency parameter and the Mach number. To the best of our knowledge, this is the first work that proves (almost) global convergence for large and ill-prepared initial data in the low Mach number limit for compressible two-fluid model in critical framework.