Vanishing capillarity limit of a generic compressible two-fluid model with common pressure

This paper concerns vanishing capillarity limit problem of a generic compressible two-fluid model with common pressure (P+ = P - ) in R 3 . Due to partial dissipation property of the system and strong coupling effects between two fluids, up to now, the vanishing capillarity limit of the 3D compressible two-fluid model with common pressure has remained a challenging problem. By exploiting the intrinsic dissipation structure of the model and employing several key observations, we show that the unique smooth solution of the generic compressible two-fluid model exists for all time, and converges globally in time to unique smooth solution of the compressible two-fluid Navier-Stokes equations, as the capillary coefficient a tends to zero. Moreover, as a by-product, we also obtain the convergence rate estimates with respect the capillary coefficient a for any given positive time. To the best of our knowledge, we establish the first result on the vanishing capillarity limit of the 3D compressible two-fluid model with common pressure. The method relies upon exploitation of the dissipation structure of the nonlinear system, the choice of auxiliary density N + = rho - (alpha + rho + - 1 ) + rho + (alpha - rho - - 1 ) , which has better regularity than ones of fraction densities (alpha +/- rho +/- ) themselves, and the introduction of a new quantity F := ( 2 mu + + )+) div u + - ( 2 mu - ) - ) div u - + a (rho+An+ - rho-An-) which plays an important role in closing energy estimates of fraction densities (alpha +/-rho +/-).