Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations

摘要

For the class of cylindrically symmetric velocity fields U(r, z, t) = {u(r, t), v(r, t), zgamma(r, t)}, two infinite energy exact solutions of the three-dimensional incompressible Euler equations are exhibited that blow up at every point in space in finite time. The first solution is embedded within the second as a special case and in both cases upsilon = 0. Both solutions represent three-dimensional vortices which take the form of hollow cylinders for which the vorticity vector is omega = (0, omega(theta), 0). An analysis on characteristics shows how more general solutions can be constructed and analysed. [References: 15]