Abstract
We perform asymptotic analysis for the Euler–Riesz system posed in either \({\mathbb {T}}^d\) or \({\mathbb {R}}^d\) in the high-force regime and establish a quantified relaxation limit result from the Euler–Riesz system to the fractional porous medium equation. We provide a unified approach for asymptotic analysis regardless of the presence of pressure in the case of repulsive Riesz interactions, based on the modulated energy estimates, the Wasserstein distance of order 2, and the bounded Lipschitz distance. For the attractive Riesz interaction case, we consider the periodic domain and estimate a lower bound on the modulated internal energy to handle the modulated interaction energy.
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Notes
Our previous work Choi and Jeong suggests that the Cauchy problem for the system (1.1) is ill-posed for the pressureless and attractive case, i.e., \(c_K >0\) and \(c_P = 0\).
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Acknowledgements
The authors would like to thank the anonymous reviewers for the invaluable comments which significantly improved the quality and readability of this article. YPC has been supported by NRF grant (No. 2017R1C1B2012918) and Yonsei University Research Fund of 2019-22-0212 and 2020-22-0505. IJJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant (No. 2019R1F1A1058486).
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Communicated by Pierre Degond.
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Choi, YP., Jeong, IJ. Relaxation to Fractional Porous Medium Equation from Euler–Riesz System. J Nonlinear Sci 31, 95 (2021). https://doi.org/10.1007/s00332-021-09754-w
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DOI: https://doi.org/10.1007/s00332-021-09754-w