Generalized variational principles

In [7] Weinan E, Y. G. Rykov, and Y. G. Sinai have introduced a generalized variational principles in order to give a weak solution of the pressureless gas equations with initial velocity u(0) and distribution of masses given by a probability measure P. The aim of this work is to connect these generalized variational principles at each time t > 0 with the convex hull of any primitive of the map m is an element of (0, 1) --> F(-1)(m) + tu(0)(F(-1)(m)). Here F is the distribution function of the probability measure P and F(-1) is its inverse. The latter convex hull is also used to obtain the solutions of the scalar conservation law and the Hamilton-Jacobi equation associated with the pressureless gas equations.