Global well-posedness and large-time behavior of classical solutions to the Euler-Navier-Stokes system in R^3
arxiv(2024)
摘要
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 Navier-Stokes equations, we properly
use the Hodge decomposition, spectral analysis, and energy method to obtain the
L^2 time decay rates of the solution when the initial perturbation belongs to
L^1 space. Furthermore, we show that the above time decay rates are optimal.
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