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

TL;DR

This paper proves the global existence of weak solutions for 1D dissipative transport equations with nonlocal velocity fields, considering different nonlocal operators and initial data conditions based on the dissipation parameter.

Contribution

It establishes new global existence results for weak solutions of dissipative transport equations with nonlocal velocities, covering various ranges of the dissipation parameter and initial data spaces.

Findings

Global weak solutions exist for 0<γ<1 with finite energy initial data.

Weak solutions also exist for γ in (0,2) with infinite energy initial data in weighted spaces.

Results depend on the range of γ and the type of nonlocal operator used.

Abstract

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