Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations

Kyungkeun Kang; Kyunghwa Kim; Changwook Yoon

doi:10.1016/j.jmaa.2019.123750

Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations

Kyungkeun Kang, Kyunghwa Kim, Changwook Yoon

Department of Mathematics

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

5 Citations (Scopus)

Abstract

We establish the global well-posedness for the following chemotaxis-fluid system {∂_tn+u⋅∇n=Δn−∇⋅(n∇c)−μn^q,∂_tc+u⋅∇c=Δc−c+n,∂_tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in R^d, d=2,3, where μ>0, q>2− [Formula presented] and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2− [Formula presented] and κ=0, very weak solutions are constructed as well.

Original language	English
Article number	123750
Journal	Journal of Mathematical Analysis and Applications
Volume	485
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2019.123750
Publication status	Published - 2020 May 1

Bibliographical note

Funding Information:
Authors greatly appreciate valuable comments and suggestions of the anonymous reviewer of this paper. Kyungkeun Kang's work is partially supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (NRF-2017R1A2B4006484 and NRF-2015R1A5A1009350). Changwook Yoon was supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (NRF-2017R1D1A1B03031035 and NRF-2017R1E1A1A03070652).

Publisher Copyright:
© 2019 Elsevier Inc.

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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title = "Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations",

abstract = "We establish the global well-posedness for the following chemotaxis-fluid system {∂tn+u⋅∇n=Δn−∇⋅(n∇c)−μnq,∂tc+u⋅∇c=Δc−c+n,∂tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in Rd, d=2,3, where μ>0, q>2− [Formula presented] and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2− [Formula presented] and κ=0, very weak solutions are constructed as well.",

author = "Kyungkeun Kang and Kyunghwa Kim and Changwook Yoon",

note = "Funding Information: Authors greatly appreciate valuable comments and suggestions of the anonymous reviewer of this paper. Kyungkeun Kang's work is partially supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (NRF-2017R1A2B4006484 and NRF-2015R1A5A1009350). Changwook Yoon was supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (NRF-2017R1D1A1B03031035 and NRF-2017R1E1A1A03070652). Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

year = "2020",

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doi = "10.1016/j.jmaa.2019.123750",

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AU - Yoon, Changwook

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PY - 2020/5/1

Y1 - 2020/5/1

N2 - We establish the global well-posedness for the following chemotaxis-fluid system {∂tn+u⋅∇n=Δn−∇⋅(n∇c)−μnq,∂tc+u⋅∇c=Δc−c+n,∂tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in Rd, d=2,3, where μ>0, q>2− [Formula presented] and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2− [Formula presented] and κ=0, very weak solutions are constructed as well.

AB - We establish the global well-posedness for the following chemotaxis-fluid system {∂tn+u⋅∇n=Δn−∇⋅(n∇c)−μnq,∂tc+u⋅∇c=Δc−c+n,∂tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in Rd, d=2,3, where μ>0, q>2− [Formula presented] and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2− [Formula presented] and κ=0, very weak solutions are constructed as well.

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Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations

Abstract

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