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

Hantaek Bae; Rafael Granero-Belinchón; Omar Lazar; Hantaek Bae; Rafael Granero-Belinchón; Omar Lazar

doi:10.3934/nhm.2019019

Networks and Heterogeneous Media

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

Received: 01 June 2018 Revised: 01 February 2019
Primary: 35A01; Secondary: 35D30, 35Q35, 35Q86

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.
- Fluid equations,
- 1D models,
- global weak solution
Citation: Hantaek Bae, Rafael Granero-Belinchón, Omar Lazar. On the local and global existence of solutions to 1d transport equations with nonlocal velocity[J]. Networks and Heterogeneous Media, 2019, 14(3): 471-487. doi: 10.3934/nhm.2019019

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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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