The problem of regulating air traffic in the en route airspace of the National Airspace System is studied using an Eulerian network model to describe air traffic flow. The evolution of traffic on each edge of the network is modeled by a modified Lighthill-Whitham-Richards partial differential equation. We pose the problem of optimal traffic flow regulation as a continuous optimization program in which the partial differential equation appears in the constraints. The equation is transformed with a variable change which removes the nonlinearity in the control variables and enables us to use linear finite difference schemes to discretize the problem. Corresponding linear programming and quadratic programming based solutions to this convex optimization program yield a globally optimal solution. The technique is applied for a network scenario in the Oakland Air Route Traffic Control Center.