Compressible Euler Equations with Vacuum

In this paper, we first show that the regular solutions of compressible Euler equations inR3with damping will not be global if the initial density function has compact support. This implies thatc2cannot be smooth across the boundaryΓseparating the gas and the vacuum after a finite time, wherecis the speed of sound. Then we study the local existence of solutions for isentropic gas flow inRwhencα, 0<α 1, is smooth acrossΓ, using the energy method and the characteristics method.