The Euler Equations and Nonlocal Conservative Riccati Equations

This note presents an infinite-dimensional family of exact solutions of the in-compressible three-dimensional Euler equations ∂tu+ u · ∇u+∇p = 0, ∇ · u = 0. (1) The solutions we present have infinite kinetic energy and blow-up in finite time. Blow-up of other similar infinite energy solutions of Euler equations has been proved before in [7] , [2]. The particular type of solution we describe was proposed in [4]. The Eulerian-Lagrangian approach we take in [3] is not restricted to this particular case, but exact integration of the equations is.We consider a two-dimensional basic squareQ of side L. The particular form of the solutions in [4] is u(x, y, z, t) = u(x, y, t), zγ(x, y, t