Global well-posedness and large-time behavior of classical solutions to the Euler-Navier-Stokes system in R 3

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible NavierStokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L2 time decay rates of the solution when the initial perturbation belongs to L1 space. Furthermore, we show that the above time decay rates are optimal. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.