The Initial Value Problem (IVP) is known to be solvable by convex minimization, at least locally in time, for the class of hyperbolic systems of conservation laws enjoying a convex entropy. This talk is rather devoted to parabolic problems. We first show that solutions can be recovered in a similar way but for arbitrarily long time intervals in simple cases such as the porous medium equation and the viscous Hamilton-Jacobi equation. For the Navier-Stokes equations, we focus on the resulting convex minimization problem and show its connection with the concept of matrix-valued optimal transport.