Optimal Large-time Behavior Of The Two-phase Fluid System In The Whole Space

In this paper,we study the large-time behavior of the classical solution of the two phase fluid model in the whole space.The model was first derived by Choi[1]by employing the hydrodynamic limit of the Vlasov-Fokker-Planck isentropic Navier-Stokes equation with strong local alignment forces.Under the assumption that the initial perturbation around the equilibrium state is sufficiently small,the literature[1]establishes the overall fitness problem.However,as Choi[1]points out in the literature,the large-time behavior of these solutions remains an open problem.In this paper,we solve this problem,proving that the heat equation converges to its associated equilibrium with the same optimal rate.In particular,the optimal decay rate of the known higher-order spatial derivatives is also obtained.Furthermore,for carefully selected data,we also show a lower bound on the rate of decay.Our method is based on Hodge decomposition,low and high frequency decomposition,spectral analysis and energy methods.