On the nonuniqueness of weak solution of the Euler equation

Weak solution of the Euler equations is defined as an L2-vector field satisfying the integral relations expressing the mass and momentum balance. Their general nature has been quite unclear. In this work an example of a weak solution on a 2-dimensional torus is constructed that is identically zero outside a finite time interval. This example is simpler and more transparent than the previous example of V. Scheffer (J. Geom. Anal. 3(4), 1993, pp. 343–401).