Potentially singular solutions of the 3D axisymmetric Euler equations

Abstract

The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(ts- t)-2.46, where ts∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ2= 0.003505 at which time the vorticity amplifies by more than (3 × 108)-fold and the maximum mesh resolution exceeds (3 × 1012)2. The vorticity vector is observed to maintain four significant digits throughout the computations.