Formation of singularities of spherically symmetric solutions to the 3D compressible Euler equations and Euler–Poisson equations

Abstract

By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the compact support assumptions on the initial data in Makino and Perthame (Jpn J Appl Math, 1990) and Perthame (Jpn J Appl Math, 1990). Our proof also simplifies that of Lei et al. (Math Res Lett, 2013) for the Euler Equations and is undifferentiated in dimensions.