Abstract: We consider 1D dissipative transport equations with nonlocal velocity field: ?t + u?x + ?ux? + ?? ? = 0, u = N (?), where N is a nonlocal operator given by a Fourier multiplier. We especially consider two types of nonlocal operators: (1) N = ?, the Hilbert transform, (2) N = (1-?xx)-?. In this paper, we show several global existence of weak solutions depending on the range of ?, ? and ?. When 0 < ? < 1, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when ? ? (0, 2). © 2018 IOP Publishing Ltd & London Mathematical Society.

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