Singularity formation for compressible Euler equations with time-dependent damping

Abstract

In this paper, we consider the compressible Euler equations with time-dependent damping α (1+t) λ u in one space dimension. By constructing "decoupled" 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 λ = 1 and λ = 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 < γ < 3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.

Global existence and stability of time-periodic solution to isentropic compressible Euler equations with source term

“…Because of the widespread application background, the compressible Euler equation with several kinds of source term have been studied extensively and there are fruitful results. For example, we can refer [6,7,16,21] for the research on the existence and stability of the small smooth solution, [2,3,5,8,9,23,26] for the singularity formation of smooth solution and the results on weak solution. In this paper, we are interested in the time-periodic solution of problem (1.1)- (1.3).…”

Time-periodic solution to nonhomogeneous isentropic compressible Euler equations with time-periodic boundary conditions

“…Intensive literatures have investigated the isentropic compressible Euler system with source terms. We refer to [3,22] and the references therein for the results on the formation of singularity, [2,5,9,10,26] for the existence and large time behavior of weak solutions, and [3,4,7,8,12,16,18] for the asymptotic behavior of smooth solutions, etc.…”