A moment approach for entropy solutions to scalar conservation laws

In this talk, a new numerical scheme, based on the moment-Sum Of Squares (SOS) hierarchy, a.k.a. Lasserre hierarchy, to solve scalar conservation laws is presented. The approach is based on a very weak notion of solution due to Di Perna, called entropy measure-valued solution. Among other nice properties, this formulation is linear with respect to a Borel measure - the measure valued solution -, which is the unknown of the equation. Moreover, it is equivalent to the celebrated entropy formulation. The aim of this seminar is to explain that the Lasserre hierarchy allows to solve such a linear equation without relying on time/space discretization, but rather by truncating the moments of the measure under consideration up to a certain degree. Approximations of the moments of the solution is, then, obtained. Another point to be discussed in the talk is the reconstruction of the graph of the solution, based on some moments data. This latter technique is based on the introduction of a suitable Christoffel-Darboux polynomial. This talk is based on some recent results obtained jointly with Didier Henrion, Jean Bernard Lasserre, Edouard Pauwels and Tillmann Weisser.