  The Geometric Thin-Film equation is a mathematical model of droplet spreading
in the long-wave limit, which includes a regularization of the contact-line
singularity. We show that the weak formulation of the problem, given initial
Radon data, admits solutions that are globally defined for all time and are
expressible as push-forwards of Borel measurable functions whose behaviour is
governed by a set of ordinary differential equations (ODEs). The existence is
first demonstrated in the special case of a finite weighted sum of delta
functions whose centres evolve over time -- these are known as `particle
solutions'. In the general case, we construct a convergent sequence of particle
solutions whose limit yields a solution of the above form. Moreover, we
demonstrate that all weak solutions constructed in this way are $1/2$-H\"older
continuous in time and are uniquely determined by the initial conditions.
