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

Abstract

We consider 1D equations with nonlocal velocity of the form

w(t) + uw(x) delta u(x)w = -v Lambda(gamma)w

where the nonlocal velocity u is given by (1) u = (1-partial derivative(xx))-(beta)w, beta > 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.