Global existence of weak solutions to dissipative transport equations with nonlocal velocity

Abstract

We consider 1D dissipative transport equations with nonlocal velocity field: where is a nonlocal operator given by a Fourier multiplier. We especially consider two types of nonlocal operators:(1) , the Hilbert transform,(2) .In this paper, we show several global existence of weak solutions depending on the range of γ, δ and α. When , 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 .