On the local and global existence of solutions to 1d transport equations with nonlocal velocity

Abstract

We consider the 1D transport equation with nonlocal velocity field: \begin{document}$ \begin{equation*} \label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta = 0, & u = \mathcal{N}(\theta), \end{split} \end{equation*} $\end{document} where \begin{document}$ \mathcal{N} $\end{document} is a nonlocal operator and \begin{document}$ \Lambda^{\gamma} $\end{document} is a Fourier multiplier defined by \begin{document}$ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $\end{document} . In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.