  We establish convergence in the diffusive limit from entropy weak solutions
of the equations of compressible gas dynamics with friction to the porous media
equation away from vacuum. The result is based on a Lyapunov type of functional
provided by a calculation of the relative entropy. The relative entropy method
is also employed to establish convergence from entropic weak solutions of
viscoelasticity with memory to the system of viscoelasticity of the rate-type.
