2017/11/08 by Elgindi, Tarek M., Jeong, In-Jee
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.03089
This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric 3D Euler equation in the domains \(x1,x2,x3) ∈ ℝ3 : x32 ≤ \mathfrakc(x12 + x22) \ for \mathfrakc sufficiently small in a scale of critical spaces. Second, we will prove that if the vorticity at t=0 can be decomposed into a scale-invariant part and a smoother part vanishing at x=0, then this decomposition remains valid for t>0 so long as the solution exists. This will then immediately imply singularity formation for finite-energy solutions in those critical spaces using that there are scale-invariant solutions which break down in finite time (this was proved in the companion paper [EJE1]). Third, we establish global regularity in the same domain and in the same scale of critical spaces for 0-swirl solutions to the system. This implies that the finite-time singularity is not coming from the domain or the critical regularity of the data (though, these are important for the present construction), they are coming from a genuine vorticity growth mechanism due to the presence of the swirl. This work and the companion work seem to be the first case when this mechanism has been effectively used to establish finite-time singularity formation for the actual 3D Euler system.