The Euler equations and nonlocal conservative Riccati equations. (English) Zbl 0970.76017
The author constructs exact solutions for three-dimensional Euler equations. The solutions are considered in the form \((u_1(x,y,t), u_2(x,y,t), z\gamma(x,y,t))\) with scalar function \(\gamma\) satisfying a nonlocal Riccati equation. It is shown that there exist smooth initial data for which periodic in \(x,y\) solutions have infinite kinetic energy and blow-up at finite time.
Reviewer: Igor V.Skrypnik (Donetsk)
MSC:
76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
35Q35 | PDEs in connection with fluid mechanics |