Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system

Hirata, Misaki, et al. "Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 11-20. <http://eudml.org/doc/294922>.

@inProceedings{Hirata2017,
abstract = {This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics \begin\{align*\} \{\left\lbrace \begin\{array\}\{ll\} (1)\_t+u\cdot 1 =\mathbb \{D\}1-\chi \_1\cdot (1c)+\mu \_11(1-1-a\_12) &\text\{in\}\ \times (0,\infty ), \\ (2)\_t+u\cdot 2 =\mathbb \{D\}2-\chi \_2\cdot (2c)+\mu \_22(1-a\_21-2) &\text\{in\}\ \times (0,\infty ), \\ c\_t+u\cdot c =\mathbb \{D\}c-(\alpha 1+\beta 2)c &\text\{in\}\ \times (0,\infty ), \\ u\_t+(u\cdot )u =\mathbb \{D\}u+\nabla P+(\gamma 1+2)\Phi , \quad \cdot u=0 &\text\{in\}\ \times (0,\infty ) \end\{array\}\right.\} \end\{align*\} under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.},
author = {Hirata, Misaki, Kurima, Shunsuke, Mizukami, Masaaki, Yokota, Tomomi},
booktitle = {Proceedings of Equadiff 14},
keywords = {Chemotaxis, Navier–Stokes, Lotka–Volterra, large-time behaviour},
location = {Bratislava},
pages = {11-20},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system},
url = {http://eudml.org/doc/294922},
year = {2017},
}

TY - CLSWK
AU - Hirata, Misaki
AU - Kurima, Shunsuke
AU - Mizukami, Masaaki
AU - Yokota, Tomomi
TI - Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 11
EP - 20
AB - This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics \begin{align*} {\left\lbrace \begin{array}{ll} (1)_t+u\cdot 1 =\mathbb {D}1-\chi _1\cdot (1c)+\mu _11(1-1-a_12) &\text{in}\ \times (0,\infty ), \\ (2)_t+u\cdot 2 =\mathbb {D}2-\chi _2\cdot (2c)+\mu _22(1-a_21-2) &\text{in}\ \times (0,\infty ), \\ c_t+u\cdot c =\mathbb {D}c-(\alpha 1+\beta 2)c &\text{in}\ \times (0,\infty ), \\ u_t+(u\cdot )u =\mathbb {D}u+\nabla P+(\gamma 1+2)\Phi , \quad \cdot u=0 &\text{in}\ \times (0,\infty ) \end{array}\right.} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.
KW - Chemotaxis, Navier–Stokes, Lotka–Volterra, large-time behaviour
UR - http://eudml.org/doc/294922
ER -