  We propose to solve polynomial hyperbolic partial differential equations
(PDEs) with convex optimization. This approach is based on a very weak notion
of solution of the nonlinear equation, namely the measure-valued (mv) solution,
satisfying a linear equation in the space of Borel measures. The aim of this
paper is, first, to provide the conditions that ensure the equivalence between
the two formulations and, second, to introduce a method which approximates the
infinite-dimensional linear problem by a hierarchy of convex,
finite-dimensional, semidefinite programming problems. This result is then
illustrated on the celebrated Burgers equation. We also compare our results
with an existing numerical scheme, namely the Godunov scheme.
