||In this talk, we will discuss the regularity or the possible
loss of regularity of solutions to the Burgers's equation or the quasi-geostrophic equation with diffusion terms given by fractional Laplacians. In the talk, we will begin with two different characterizations of the critical fractional Laplacian with exponent 1/2 to motivate the talk. In particular, we would like to stress the nonlocal nature of the fractional Laplacian, and to say something about the way in
which the solutions are affected by the presence of a fractional Laplacian as a nonlocal operator in the PDE in question. In particular, we will bring to the audiences'attention the recent important works in this area
due to L. Caffarelli, A. Vasseur, A. Kiselev, F. Nazarov etc.