A moment approach for entropy solutions to scalar conservation laws

In this talk, we will propose a new numerical scheme, based on the moment-SOS hierarchy, a.k.a. Lasserre hierarchy, to solve scalar conservation laws. Our approach is based on a very weak notion of solution due to Di Perna, which is called entropy measure-valued solution. Among other nice properties, this formulation is linear in a Borel measure - the measure valued solution -, which is the unknown of the equation, and moreover it is equivalent to the well-known entropy solution formulation. Our aim 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. We then obtains approximations of the moments of the solution. We will also explain how to reconstruct the graph of the solution, based on some moments data. This talk is based on some recent results obtained jointly with Didier Henrion, Jean Bernard Lasserre, Edouard Pauwels and Tillmann Weisser.