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

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 $\gamma \in (0,2)$.