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

Abstract

We consider 1D dissipative transport equations with nonlocal velocity field: \[ θ_t+uθ_x+δu_{x} θ+Λ^γθ=0, \quad u=\mathcal{N}(θ), \] 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 $\gamma \in (0,2)$.

28 pages, improved and extended version (some extra assumptions have been removed, new cases are treated)