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, one can find, in the fluid mechanics literature, examples of topology-preserving diffusion equations. They are very degenerate since they admit all stationary solutions to the Euler equations as equilibrium points. For them, we provide a suitable concept of "dissipative gradient-flow solutions", which shares common features both with the dissipative solutions of P.-L. Lions for the Euler equations and the gradient-flow solutions "a la De Giorgi" recently used by Ambrosio-Gigli-Savare for the scalar heat equation in very general metric spaces. We show that the initial value problem admits such global solutions and they are unique whenever they are smooth.