One of the important notions of quasi-local mass in general relativity is the one proposed by Hawking in 1968, nowadays commonly known as the Hawking mass. In this talk, we study the L2-gradient flow of the Hawking mass functional on a closed surface in the Riemannian Schwarzschild 3-manifold. We begin by a brief discussion of the higher order estimates, to see that the uniform curvature bounds hold under the absence of curvature concentration. Then, we carry out a blowup analysis to determine the required condition in order to eliminate such concentration phenomenon. We focus on the comparison between our work and the Willmore flow on a closed surface in R3. Finally, we conclude by establishing the longtime existence of the solution.