Compressible Euler equations interacting with incompressible flow

We investigate the global existence and large-time behavior of clas- sical solutions to the compressible Euler equations coupled to the incompress- ible Navier-Stokes equations. The coupled hydrodynamic equations are rigor- ously derived in [1] as the hydrodynamic limit of the Vlasov/incompressible Navier-Stokes system with strong noise and local alignment. We prove the existence and uniqueness of global classical solutions of the coupled system under suitable assumptions. As a direct consequence of our result, we can con- clude that the estimates of hydrodynamic limit studied in [1] hold for all time. For the large-time behavior of the classical solutions, we show that two fluid velocities will be aligned with each other exponentially fast as time evolves.