In this talk, we will discuss the convergence of the Ohta-Kawasaki equations to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions N =2, 3. Roughly speaking, this motion describes an interface moving with velocity equal to the surface Laplacian of its mean curvature. These equations arise in modeling microphase separation in diblock copolymers. For the case of radially symmetric initial data without well- preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Along the way, we prove an $H^{1}$-version of De Giorgi's conjecture in space dimensions N=2, 3. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.