The singularities for a periodic transport equation

Abstract

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$u_t-(Hu{)}_x{u}_x+\kappa {\Lambda }^{\alpha }u=0,(t,x)\in R^{+}\times S,$$ where $\kappa \ge 0$, $0<\alpha \le 1$ 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 $\kappa {\Lambda }^{\alpha }u$, we also prove that the finite time blowup would occur.