Existence of weak solutions in wasserstein space for a chemotaxis model coupled to fluid equations

Kyungkeun Kang; Hwa Kil Kim

doi:10.1137/16M1083232

Existence of weak solutions in wasserstein space for a chemotaxis model coupled to fluid equations

Kyungkeun Kang, Hwa Kil Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider a coupled system of Keller-Segel-type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. We first establish the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has finite entropy. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in the case that its initial mass is assumed to be bounded and integrable.

Original language	English
Pages (from-to)	2965-3004
Number of pages	40
Journal	SIAM Journal on Mathematical Analysis
Volume	49
Issue number	4
DOIs	https://doi.org/10.1137/16M1083232
Publication status	Published - 2017

Bibliographical note

Funding Information:
∗Received by the editors July 5, 2016; accepted for publication (in revised form) May 2, 2017; published electronically August 3, 2017. http://www.siam.org/journals/sima/49-4/M108323.html Funding: The work of the first author was supported by NRF-2014R1A2A1A11051161 and NRF-20151009350. The work of the second author was supported by NRF-2015R1D1A1A01056696. †Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea (kkang@yonsei. ac.kr). ‡School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea (hwakil@kias.re.kr).

Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

All Science Journal Classification (ASJC) codes

Analysis
Computational Mathematics
Applied Mathematics

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title = "Existence of weak solutions in wasserstein space for a chemotaxis model coupled to fluid equations",

abstract = "We consider a coupled system of Keller-Segel-type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. We first establish the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has finite entropy. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in the case that its initial mass is assumed to be bounded and integrable.",

author = "Kyungkeun Kang and Kim, {Hwa Kil}",

note = "Funding Information: ∗Received by the editors July 5, 2016; accepted for publication (in revised form) May 2, 2017; published electronically August 3, 2017. http://www.siam.org/journals/sima/49-4/M108323.html Funding: The work of the first author was supported by NRF-2014R1A2A1A11051161 and NRF-20151009350. The work of the second author was supported by NRF-2015R1D1A1A01056696. †Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea (kkang@yonsei. ac.kr). ‡School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea (hwakil@kias.re.kr). Publisher Copyright: {\textcopyright} 2017 Society for Industrial and Applied Mathematics.",

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N2 - We consider a coupled system of Keller-Segel-type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. We first establish the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has finite entropy. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in the case that its initial mass is assumed to be bounded and integrable.

AB - We consider a coupled system of Keller-Segel-type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. We first establish the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has finite entropy. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in the case that its initial mass is assumed to be bounded and integrable.

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Existence of weak solutions in wasserstein space for a chemotaxis model coupled to fluid equations

Abstract

Bibliographical note

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