The Cauchy problem of a pressureless two-phase flow system in three-dimensional space

In this paper, we study a two-phase flow system consisting of the compressible pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force 𝜌 ⁢ ( 𝑢 − 𝑣 ) . Due to the absence of pressure term in the Euler equations, the classical existence theory of the hyperbolic-parabolic system developed by Kawashima in [Doctoral Dissertation, Kyoto University, 1984] is not applicable to this system. Therefore, a new approach has been introduced here to address this issue. Firstly, we employ the pure energy method to derive a faster time-decay rate for the velocity of Euler flow ∥ 𝛻 2 𝑢 ∥ 𝐻 2 ≤ 𝐶 ⁢ ( 1 + 𝑡 ) − 2 + 𝜎 2 under the assumption that initial velocity of Navier-Stokes flow 𝑣 0 ∈ ˙ 𝐻 − 𝜎 ⁡ ( ℝ 3 ) for some 𝜎 ∈ ( 0 , 3 2 ) and initial density ρ 0 > 0. Then, combining the above decay estimate and characteristic method yields the uniform boundedness of the density ρ . Finally, utilizing the uniform estimates of velocities and the continuity argument, we establish the global well-posedness of this two-phase flow system. As a by-product, it is also shown that the velocities ( u, v ) decay to the motionless state at the optimal algebraic time-decay rates in L 2 -norm. Our results reveal that the smoothing effect of the Navier-Stokes equations can be propagated to Euler equations through the drag force.