The Cauchy Problem Of The Unstressed Euler-Navier-Stokes Equation With Degenerate Damping

In this paper,we consider the global existence and uniqueness of the classical solution to the Cauchy problem of the following pressureless Euler-Navier-Stokes equations with degenerate damping in the whole space:(?)where d?3,?=?(x,t)and u=u(x,t)represent the fluid density and velocity for the pressureless flow,respectively,and v=v(x,t)denotes the fluid velocity for the incompressible flow,?>0 is the viscosity coefficient,m>0,??0 are constants in the coefficients of friction term m/(1+t)?.By the long time behavior estimation of the solution to the heat equation,the total energy time decay estimate is obtained.Together with a priori estimation of the solution in time-weighted Sobolev space and the bootstrapping argument,we establish the global existence and uniqueness of the classical solution(?,u,v)if 0???1 and when the initial data are sufficiently small and regular.