Global existence, singularities and ill-posedness for a nonlocal flux

In this paper we study a one-dimensional model equation with a nonlocal flux given by the Hilbert transform that is related with the complex inviscid Burgers equation. This equation arises in different contexts to characterize nonlocal and nonlinear behaviors. We show global existence, local existence, blow-up in finite time and ill-posedness depending on the sign of the initial data for classical solutions.