Finite-time singularity formation for C¹ solutions to the compressible Euler equations with time-dependent damping

In this paper, the initial-boundary value problem of the multidimensional compressible Euler equations with time-dependent damping in radial symmetry is considered. It is shown that finite-time singularity will be developed for the C¹ solutions of the compressible Euler equations with time-dependent damping coefficients μ/((1+t)^λ), μ > 0, λ ≥ 0 if the initial value of a newly introduced functional, F(t) = ∫₁^∞ ρ(t,r)u(t,r) [r^N − α(t)] dr with a time-dependent parameter α(t) is sufficiently large. The blowup conditions imply that the initial kinetic energy of the fluid must not be less than a given constant. Copyright © 2019 Informa UK Limited, trading as Taylor & Francis Group.