Relaxation Limit in Besov Spaces for Compressible Euler Equations

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.