The singularities for a periodic transport equation

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+kappaLambda^{alpha}u=0,quad (t,x)in R^{+}times S, $$ where $kappageq0$, $0<alphaleq1$ and $S=[-pi,pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $kappaLambda^{alpha}u$, we also prove that the finite time blowup would occur.