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

Abstract

We consider 1D dissipative transport equations with nonlocal velocity field: theta(t) + u theta(x) + delta u(x)theta + Lambda(gamma)theta = 0, u = N(theta), where N is a nonlocal operator given by a Fourier multiplier. We especially consider two types of nonlocal operators: (1) N = H, the Hilbert transform, (2) N = (1 - partial derivative(xx))-(alpha). In this paper, we show several global existence of weak solutions depending on the range of gamma, delta and alpha. When 0 < gamma < 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 gamma is an element of(0, 2).