Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations

The Euler-Navier-Stokes (E-NS) system arises as a macroscopic description of kinetic-fluid interactions, derived from the local-Maxwellian closure of the Vlasov-Fokker-Planck-Navier-Stokes flow. In this paper, we investigate the singular limit of the system in $\mathbb{R}^d$ ($d\ge2$) when the relaxation parameter $\varepsilon>0$ tends to zero. In contrast to the Euler system with velocity damping, the E-NS model features only a weaker relaxation of the relative velocity, which makes it challenging to analyze its dynamics as $\varepsilon\rightarrow 0$. We develop an energy argument to show global-in-time error estimates between the E-NS system and its limit system, the so-called Kramers-Smoluchowski-Navier-Stokes (KS-NS) system. These error estimates enable us to prove the global existence and uniform-in-$\varepsilon$ regularity of the strong solution to the E-NS system in a hybrid critical Besov space with a sharp frequency threshold of order $\mathcal{O}(\varepsilon^{-1})$ separating the low- and high-frequency regimes. Moreover, the large-time asymptotic stability of the global solution to the E-NS system is established. More precisely, we derive the optimal decay rates of the solution uniformly in $\varepsilon$, and the enhanced decay rates for the difference between the densities of the E-NS system and the KS-NS system.