Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u), x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.