Initial value problems viewed as generalized optimal transport problems with matrix-valued density fields

The initial value problem for many important PDEs (Burgers, Euler, Hamilton-Jacobi, Navier-Stokes equations, systems of conservation laws with convex entropy, etc…) can be often reduced to a convex minimization problem that can be seen as a generalized optimal transport problem involving matrix-valued density fields. The time boundary conditions enjoy a backward-forward structure of “ballistic” type, just as in mean-field game theory.