  The relaxation limit in critical Besov spaces for the multidimensional
compressible Euler equations is considered. As the first step of this
justification, the uniform (global) classical solutions to the Cauchy problem
with initial data close to an equilibrium state are constructed in the
Chemin-Lerner's spaces with critical regularity. Furthermore, it is shown that
the density converges towards the solution to the porous medium equation, as
the relaxation time tends to zero. Several important estimates are achieved,
including a crucial estimate of commutator.
