The singularities for a periodic transport equation

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation ut − (Hu)xux + κΛu = 0, (t, x) ∈ R × S, where κ ≥ 0, 0 < α ≤ 1 and S = [−π, π]. We first establish the local-in-time well-posedness for this transport equation in H(S). In the case of κ = 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 κΛu, we also prove that the finite time blowup would occur.