Low Mach Number Limit and Convergence Rates for a Compressible Two-Fluid Model with Algebraic Pressure Closure

TL;DR

This paper rigorously analyzes the low Mach number limit for a viscous compressible two-fluid model with algebraic pressure closure, proving convergence to incompressible Navier-Stokes equations with explicit rates.

Contribution

It establishes the convergence and rates for the low Mach number limit in a two-fluid model with implicit pressure laws, a more delicate scenario than explicit pressure models.

Findings

Solutions exist uniformly in Mach number for well-prepared data.

Solutions converge to incompressible Navier-Stokes solutions as Mach number approaches zero.

Explicit convergence rates for densities and velocity are derived.

Abstract

We study the low Mach number limit for a viscous compressible two-fluid model with algebraic pressure closure in the three-dimensional torus $\mathbb{T}^3$. The pressure is determined implicitly through the densities of the two phases, which makes the singular limit substantially more delicate than for models with explicit pressure laws. Working in the framework of local-in-time strong solutions, we prove that, for well-prepared initial data, solutions to the rescaled compressible two-fluid system exist on a time interval independent of the Mach number and converge to the solution of the incompressible Navier--Stokes equations as the Mach number tends to zero. In addition, we establish explicit convergence rates for the densities and the velocity field. The proof relies on uniform high-order energy estimates and a relative energy argument adapted to the implicit structure of the pressure law. These results provide a rigorous justification of the low Mach number limit for the compressible two-fluid model with algebraic pressure closure.