On the well-posedness of various one-dimensional model equations for fluid motion

We consider 1D equations with nonlocal velocity of the form wt+uwx+δuxw=−νΛγwwhere the nonlocal velocity u is given by (1) u=(1−∂xx)−βw, β>0 or (2) u=Hw (H is the Hilbert transform). In this paper, we address several local well-posedness results with blow-up criteria for smooth initial data. We then establish the global well-posedness by using the blow-up criteria.