We discuss diffusion equations that are constrained to preserve the volume of each level set during the time evolution (which excludes the standard heat equation). We consider, in particular, the gradient flow of the Dirichlet integral under suitable volume-preserving transportation metrics. The resulting equations are non-linear and very degenerate, admitting as stationary solutions all scalar functions which are functions of their own Laplacian. (In particular, in 2D, all stationary solutions of the Euler equations for incompressible fluids.) We relate them to both combinatorial optimization and linear algebra, through the quadratic assignmemt problem (a NP combinatorial optimization problem including the travelling salesman problem) and the Brockett-Wegner diagonalizing flow for linear operators. For these equations, we provide a concept of "dissipative solutions" that exist globally in time and are unique as long as they stay smooth, following some works of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savare (for the heat equation in metric spaces).