The classical equations of hydrodynamics (Euler, Navier-Stokes) are of mildly non-local nature through the pressure term. Simplified models, like the Burgers equation, focus on a scalar field to remove geometrical constraints.

We present here a non-local variant of the Burgers equation, whose instabilities are tied to the local sign of the solution, along with some results obtained with R.Shvydkoy, C. Imbert, J. Tan in the positive case and with R. Anton, K. Verdure in the unsigned case.