Relative entropy method for the relaxation limit of hydrodynamic models

José Antonio Carrillo; Yingping Peng; Aneta Wróblewska-Kamińska; José Antonio Carrillo; Yingping Peng; Aneta Wróblewska-Kamińska

doi:10.3934/nhm.2020023

Networks and Heterogeneous Media

2020, Volume 15, Issue 3: 369-387. doi: 10.3934/nhm.2020023

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Relative entropy method for the relaxation limit of hydrodynamic models

1.
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
2.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China and, Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
3.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

Received: 01 October 2019 Revised: 01 April 2020 Published: 09 September 2020
Primary: 35B25, 35Q31; Secondary: 35Q92

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.
- Relative entropy method,
- aggregation-diffusions,
- nonlocal hydrodynamics,
- relaxation limit
Citation: José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models[J]. Networks and Heterogeneous Media, 2020, 15(3): 369-387. doi: 10.3934/nhm.2020023

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Abstract

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.

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