SINGULARITY FORMATION FOR COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

In this paper, we consider the compressible Euler equations with time-dependent damping alpha/(1+t)(lambda)u in one space dimension. By constructing "de coupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case lambda not equal 1 and lambda = 1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1 < gamma < 3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.