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

Hantaek Bae; Rafael Granero-Belinchón; Omar Lazar

doi:10.3934/nhm.2019019

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2019, Volume 14, Issue 3: 471-487. Doi: 10.3934/nhm.2019019

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

1.
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Republic of Korea
2.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain
3.
Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Sevilla, Spain

^* Corresponding author
^* Corresponding author

Received: May 31, 2018

Revised: January 31, 2019

Published: August 2019

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Abstract

We consider the 1D transport equation with nonlocal velocity field:

$ \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*} $

where $ \mathcal{N} $ is a nonlocal operator and $ \Lambda^{\gamma} $ is a Fourier multiplier defined by $ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.

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Mathematics Subject Classification: Primary: 35A01; Secondary: 35D30, 35Q35, 35Q86.

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