Non-critical fractional conservation laws in domains with boundary

Matthieu  Brassart; Matthieu  Brassart

doi:10.3934/nhm.2016.11.251

Networks and Heterogeneous Media

2016, Volume 11, Issue 2: 251-262. doi: 10.3934/nhm.2016.11.251

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Non-critical fractional conservation laws in domains with boundary

Matthieu Brassart ^{1
,}

1.
Laboratoire de Mathématiques de Besancon U.F.R. S.T, 16 route de Gray, 25030 BESANCON

Received: 01 May 2015 Revised: 01 July 2015
35B30, 35L65, 35L82, 35S10, 35S30.

We study bounded solutions for a multidimensional conservation law coupled with a power $s\in (0,1)$ of the Dirichlet laplacian acting in a domain. If $s \leq 1/2$ then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If $s >1/2$ then the focus is rather on the $C^\infty$-regularity of weak solutions. This kind of results is known in $\mathbb{R}^N$ but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.
- Entropy formulation,
- nonlocal conservation law,
- Kato inequality,
- $L^1$-contraction principle
Citation: Matthieu Brassart. Non-critical fractional conservation laws in domains with boundary[J]. Networks and Heterogeneous Media, 2016, 11(2): 251-262. doi: 10.3934/nhm.2016.11.251

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Abstract

We study bounded solutions for a multidimensional conservation law coupled with a power $s\in (0,1)$ of the Dirichlet laplacian acting in a domain. If $s \leq 1/2$ then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If $s >1/2$ then the focus is rather on the $C^\infty$-regularity of weak solutions. This kind of results is known in $\mathbb{R}^N$ but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.

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