# Large-friction and incompressible limits for pressureless Euler/isentropic Navier-Stokes flows

## TL;DR

This paper rigorously analyzes the limits of a coupled Euler-NS two-phase flow system under large friction and incompressibility, establishing uniform estimates, convergence rates, and asymptotic behaviors in various regimes.

## Contribution

It provides the first rigorous justification of the combined large-friction and incompressible limits for the Euler-NS system with explicit convergence rates.

## Key findings

- Established uniform regularity estimates with respect to the friction coefficient.
- Proved strong convergence of the Euler-NS system to a drift-flux model as friction vanishes.
- Analyzed the incompressible limit of the drift-flux model and the combined limit for the Euler-NS system.

## Abstract

We investigate the large-friction and incompressible limits for a two-phase flow (Euler-NS) system which couples the pressureless Euler equations and the isentropic compressible Navier-Stokes equations through a drag force term with the friction coefficient $\frac{1}{\tau}>0$ in $\mathbb{R}^{d}$ ($d\geq2$). We establish the uniform regularity estimates with respect to $\tau$ so that the solution of the Cauchy problem for the Euler-NS system exists globally in time, provided that the initial data are uniformly close to the equilibrium state in a critical Besov space. These uniform estimates allow us to rigorously justify the strong convergence of the Euler-NS system to a one-velocity two-phase drift-flux (DF) model as $\tau \to 0$, with an explicit convergence rate of order $\sqrt{\tau}$. We also study the large-time asymptotic behavior of solutions for the Euler-NS system, uniformly with respect to $\tau$. Moreover, when the Mach number $\varepsilon>0$ is considered, we prove the incompressible limit of the DF model toward the Transport-Navier-Stokes (TNS) system as $\varepsilon\rightarrow 0$, and justify the combined large-friction and incompressible limit for the Euler-NS system toward the TNS system in the regime $\tau=\varepsilon\rightarrow 0$. Each singular limit process is globally valid in time for {\emph{ill-prepared}} initial data.