  We consider 1D dissipative transport equations with nonlocal velocity field:
\[ \theta_t+u\theta_x+\delta u_{x} \theta+\Lambda^{\gamma}\theta=0, \quad
u=\mathcal{N}(\theta), \] where $\mathcal{N}$ is a nonlocal operator given by a
Fourier multiplier. Especially we consider two types of nonlocal operators:
  $\mathcal{N}=\mathcal{H}$, the Hilbert transform,
  $\mathcal{N}=(1-\partial_{xx} )^{-\alpha}$.
  In this paper, we show several global existence of weak solutions depending
on the range of $\gamma$ and $\delta$. 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 $\gamma \in (0,2)$.
