Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic...

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Bibliographic Details
Published in:Nonlinearity 2023-11, Vol.36 (11), p.5998-6019
Main Authors: Coclite, Giuseppe Maria, De Nitti, Nicola, Keimer, Alexander, Pflug, Lukas, Zuazua, Enrique
Format: Article
Language:English
Subjects:
35L65
approximation of local conservation laws
Burgers equation
entropy solutions
N-waves
nonlocal conservation laws
nonlocal flux
source-type solutions
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Summary:We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘ N -wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/acf01d