  The convergence of the Boltzmann equaiton to the compressible Euler equations
when the Knudsen number tends to zero has been a long standing open problem in
the kinetic theory. In the setting of Riemann solution that contains the
generic superposition of shock, rarefaction wave and contact discontinuity to
the Euler equations, we succeed in justifying this limit by introducing
hyperbolic waves with different solution backgrounds to capture the extra
masses carried by the hyperbolic approximation of the rarefaction wave and the
diffusion approximation of contact discontinuity.
