Regularity of solutions for the critical $N$-dimensional Burgers'   equation
  We consider the fractional Burgers' equation on $\R^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur \cite{Driftdiffusion} and show existence of smooth solutions given any initial datum in $L^2(\R^N)$. 
Chi Hin Chan, Magdalena Czubak