  We establish long-time existence for a projected Sobolev gradient flow of
generalized integral Menger curvature in the Hilbert case, and provide
$C^{1,1}$-bounds in time for the solution that only depend on the initial
curve. The self-avoidance property of integral Menger curvature guarantees that
the knot class of the initial curve is preserved under the flow, and the
projection ensures that each curve along the flow is parametrized with the same
speed as the initial configuration. Finally, we describe how to simulate this
flow numerically with substantially higher efficiency than in the corresponding
numerical $L^2$ gradient descent or other optimization methods.
