Statistical solutions of hyperbolic conservation laws

Recent theoretical and numerical results have shown that invisvid models in gas dynamics, such as the (in)compressible Euler equations, are unstable with respect to initial data or even ill-posed. Going back to the roots of turbulence theory, we interpret instead these hyperbolic conservation laws in a probabilistic manner. In this talk I will survey some recent developments in so-called statistical solutions, both theoretical and numerical. These include well-posedness for scalar conservation laws; energy conservation for regular solutions of the incompressible Euler equations; and numerical evidence for the convergence of the mean flow, structure functions etc. for compressible Euler equations.