We present a network architecture for the distributed utility max-min flow control of elastic and nonelastic flows where utility values of users (rather than data rates of users) are enforced to achieve max-min fairness. The proposed link algorithm converges to utility max-min fair bandwidth allocation in the presence of round-trip delays without using the information of users' utility functions. To show that the proposed algorithm can be stabilized not locally but globally, we found that the use of nonlinear control theory is inevitable. Even though we use a distributed flow-control algorithm, it is shown that any kind of utility function can be used as long as the minimum slopes of the functions are greater than a certain positive value. Though our analysis is limited to the single-bottleneck and homogeneous-delay case, we believe that the proposed algorithm is the first to achieve utility max-min fairness with guaranteed stability in a distributed manner.