Analysis and Numerical Solvability of Backward-Forward Conservation Laws
Abstract.
In this paper, we study the problem of initial data identification for weak-entropy solutions of the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the identification problem under consideration ill-posed. To get around this issue, we introduce a nonsmooth optimization problem, which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in \(L^2(\mathbb{R})\) norm. Here, we characterize the set of minimizers of the aforementioned nonsmooth optimization problem. One of the minimizers is the backward entropy solution, constructed using a backward-forward method. Some simulations are given using a wave-front tracking algorithm.
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Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 1949 - 1968
DOI: 10.1137/22M1478720
ISSN (online): 1095-7154
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 17 February 2022
Accepted: 5 July 2022
Published online: 9 June 2023
Keywords
MSC codes
Authors
Funding Information
Agence Nationale de la Recherche (ANR): ICON-ANR-16-ACHN-0014
Air Force Office of Scientific Research (AFOSR): FA9550-18-1-0242
H2020 European Research Council (ERC): 694126-DyCon
H2020 Marie Skłodowska-Curie Actions (MSCA): 765579-ConFlex
Eusko Jaurlaritza (Basque Government): KK-2018/00083 ROAD2DC
Ministerio de Economía, Industria y Competitividad, Gobierno de España (MINECO): MTM2017-92996-C2-1-R/2-R
Funding: The work of the authors was supported by the European Research Council (ERC) under European Union’s Horizon 2020 research and innovation programme grant 694126-DyCon, partially by ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, by MINECO grant MTM2017-92996-C2-1-R/2-R COSNET, by Air Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0242, by the Alexander von Humboldt-Professorship program, by the European Union’s Horizon 2020 research and the innovation programme under the Marie Skłodowska-Curie grant 765579-ConFlex, and by ANR grant ICON-ANR-16-ACHN-0014.
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