Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid

The Liapunov Function is a continuous function, defined in the phase space of a dynamical system, which grows along every trajectory. Its existence means certain irreversibility of the system. Generalized Liapunov Function is a continuous function with values in some topological vector space with a cone, and it grows along every trajectory in the sense of the partial order, defined by the cone. In this work we construct a Generalized Liapunov Function in the phase space of 2-dimensional ideal incompressible fluid. This function is defined in the terms of weak singularities of the flow. Likewise, we find the Generalized Integral; this is a continuous function, defined in the phase space of the fluid, assuming its values in some topological vector space, which is constant along every trajectory. This implies the existence of (a lot of) scalar valued Liapunov Functions and new integrals for the fluid. We also obtain precise result concerning propagation of singularities for the Euler equations, using the Generalized Liapunov Function.