A Nonlocal Version of Wavefront Tracking Motivated by Kuramoto-Sakaguchi Equation
In this paper, we present a modified wave-front tracking algorithm which is suitable for the analysis of scalar conservation laws with nonlocal terms. This method has been first employed in Shen and Zhang (Arch Ration Mech Anal 204:837–879, 2012) to analyze a nonlocal Hamilton-Jacobi equation related to a granular flow and later used in other works. Such an approach leads to a possibly simpler analysis in obtaining rigorous quantitative estimates on approximate solutions, compared to a classical iteration procedure based on the recomputation of the nonlocal term at each time step. Here, we delineate this method for a nonlocal equation namely “the Kuramoto-Sakaguchi equation” arising from the kinetic modeling of collective motion of large ensemble of Kuramoto oscillators, for which BV-weak solutions and their large time behavior are investigated in Amadori et al. (J Differ Equ 262, 978–1022, 2017).
The first author would like to thank the organizers of the INdAM Workshop on Innovative Algorithms and Analysis (Rome, May 17–20 2016) for their kind invitation and hospitality.
The work of D. Amadori is supported by the Miur-PRIN 2012 Project Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: theoretical and applicative aspects. The work of S.-Y. Ha is partially supported by a National Research Foundation of Korea Grant (2014R1A2A2A05002096) funded by the Korean government, and the work of J. Park was supported by NRF (National Research Foundation of Korea) Grant funded by Korean Government (NRF-2014-Fostering Core Leaders of the Future Basic Science Program).
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