Blowup for the 3D compressible Euler equations

• Comparing to Sideris’ result in 1985, a blowup result is obtained without m(0)≥0m(0)≥0.
• A blowup result is obtained without the condition η(0)≥0η(0)≥0, which is essential before.
• Several blowup results are obtained by calculating some modified functionals.

We establish several results about the blowup of the solutions for the Cauchy problem of the 3D compressible Euler equations, under some assumptions on weighed functionals associated with the initial momentum that they are large enough to develop singularities. Especially we prove the blowup results without the condition η(0)=∫[ρ(x,0)eS(x,0)γ−ρ̄eS̄γ]dx≥0, while it is an essential condition in the well-known result obtained by Thomas C. Sideris.