Recovery of hyperbolic conservation laws by space-time optimization

The principle of least action has been used for a long while to find important equations of physics and mechanics through space-time optimization. Nevertheless, it is customary to say that recovering their solutions by space-time convex minimization doesn’t make much sense. However, this is indeed the case for systems of conservation laws with a convex entropy, at least in small time, in particular the Euler equations of isothermal gases. We will discuss how this idea can be (partly) extended to Einstein’s equations in vacuum (including with a cosmological constant). We obtain formulations that are very close to those of the Euler equations (provided that density and velocity fields with matrix values are used). However, convex optimization is lost, for lack of a convex entropy and we can recover only the equations but a priori not their solutions.