The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of "dissipative solutions", which shares common features with both P.-L. Lions' dissipative solutions to the Euler equations and the concept of "curves of maximal slopes", a la De Giorgi, recently used to study the scalar heat equation in very general metric spaces.