In Cognitive Radio Networks (CRN), the appearance of more active primary users (PUs) might require blocking of one or more secondary user (SU) transmissions to maintain interference constraints at PU receivers and Quality of Service (QoS) constraints of SUs. In general, SU transmitters need to adjust their transmission powers to maintain these constraints at all times. In this paper, we are interested in maximizing the number of concurrent SU flows under new PU arrivals and performing power allocation for SU links belonging to maintained data flows to maximize the minimum flow capacity. In particular, we develop a flow maintenance and power control scheme to solve this problem, which is referred to as Flow-Based Power Control (FBPC). The proposed scheme comprises two phases where we attempt to find a best set of flows with maximum size in the first phase and maximize the minimum flow capacity through power control in the second phase. To reduce computation complexity required in the first phase, we develop a flow combination metric, which is used to find the best flow combination. Moreover, we formulate power allocation problem that maximizes the minimum flow capacity and show how to transform it into a geometric convex program. Numerical results are presented to show the efficiency of the proposed flow combination metric and the desirable performance of the proposed two-phase flow maintenance and power control scheme.