We study the global existence of a unique strong solution, and its large-time behavior, of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov-Fokker-Planck/isentropic Navier-Stokes equations with strong local alignment forces. When the initial data are sufficiently small and regular, we establish the unique existence of the global Hs-solutions in a perturbation framework. We also provide the large-time behavior of classical solutions showing the alignment between two fluid velocities exponentially fast as time evolves. For this, we construct a Lyapunov function measuring the fluctuations of momentum and mass from its averaged quantities.
Bibliographical noteFunding Information:
This work was supported by Engineering and Physical Sciences Research Council (EP/K008404/1). This work was also supported by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers.
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics