Low Mach number limit of a two-phase flow model in ${\mathbb{R}}^3$

Abstract

We study the simultaneous low Mach number limit of a compressible Navier--Stokes--Euler two-phase system in ${\mathbb{R}}^3$, in which the pressure terms in both phases are scaled by ${\epsilon }^{-2}$. For sufficiently small $H^3$-perturbations around the constant equilibrium, we establish the global well-posedness of the scaled compressible system and derive global energy-dissipation estimates that are uniform with respect to the Mach number $\epsilon$. A key feature of the analysis is the degenerate dissipation structure: viscosity acts only on the Navier--Stokes phase, while the drag coupling transfers dissipation to the inviscid Euler phase through the relaxation mode. We then prove the global well-posedness and large-time decay of the limiting incompressible two-phase system, and show that the relative velocity $u-\omega$ decays faster than the full velocity pair. Finally, for the well-prepared initial data, we introduce pressure-corrected acoustic variables to remove the singular pressure mismatch and establish a global-in-time $H^2$-error. As a result, the scaled compressible solutions converge uniformly in time to the corresponding solution of the limiting incompressible system.

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