---
_id: '63325'
abstract:
- lang: eng
text: "Abstract\r\n We consider the
spatially 2D version of the model $$\\begin{equation*} \\qquad\\quad\\left\\{
\\begin{array}{@{}rcll} n_t + u\\cdot\\nabla n &=& \\Delta n - \\nabla
\\cdot \\big(nS(x,n,c) \\cdot \\nabla c \\big), \\qquad &\\qquad x\\in \\Omega,
\\ t>0, \\\\ c_t + u\\cdot \\nabla c &=& \\Delta c - n f(c), \\qquad
&\\qquad x\\in \\Omega, \\ t>0, \\\\ u_t &=& \\Delta u + \\nabla
P + n\\nabla\\phi, \\qquad \\nabla\\cdot u=0, \\qquad &\\qquad x\\in \\Omega,
\\ t>0, \\end{array} \\right. \\qquad \\qquad (\\star) \\end{equation*}$$for
nutrient taxis processes, possibly interacting with liquid environments. Here
the particular focus is on the situation when the chemotactic sensitivity $S$
is not a scalar function but rather attains general values in ${\\mathbb{R}}^{2\\times
2}$, thus accounting for rotational flux components in accordance with experimental
findings and recent modeling approaches. Reflecting significant new challenges
that mainly stem from apparent loss of energy-like structures, especially for
initial data with large size, the knowledge on ($\\star$) so far seems essentially
restricted to results on global existence of certain generalized solutions with
possibly quite poor boundedness and regularity properties; widely unaddressed
seem aspects related to possible effects of such non-diagonal taxis mechanisms
on the qualitative solution behavior, especially with regard to the fundamental
question whether spatial structures may thereby be supported. The present work
answers the latter in the negative in the following sense: under the assumptions
that the initial data $(n_0,c_0,u_0)$ and the parameter functions $S$, $f$, and
$\\phi$ are sufficiently smooth, and that $S$ is bounded and $f$ is positive on
$(0,\\infty )$ with $f(0)=0$, it is shown that any nontrivial of these solutions
eventually becomes smooth and satisfies $$\\begin{equation*} n(\\cdot,t)\\to -
\\int_\\Omega n_0, \\quad c(\\cdot,t)\\to 0 \\quad \\text{and} \\quad u(\\cdot,t)\\to
0 \\qquad \\text{as} \\ t\\to\\infty, \\end{equation*}$$uniformly with respect
to $x\\in \\Omega$. By not requiring any smallness condition on the initial data,
the latter seems new even in the corresponding fluid-free version obtained on
letting $u\\equiv 0$ in ($\\star$)."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity
in Nutrient Taxis(-Stokes) Systems? International Mathematics Research Notices.
2019;2021(11):8106-8152. doi:10.1093/imrn/rnz056
apa: Winkler, M. (2019). Can Rotational Fluxes Impede the Tendency Toward Spatial
Homogeneity in Nutrient Taxis(-Stokes) Systems? International Mathematics Research
Notices, 2021(11), 8106–8152. https://doi.org/10.1093/imrn/rnz056
bibtex: '@article{Winkler_2019, title={Can Rotational Fluxes Impede the Tendency
Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?}, volume={2021},
DOI={10.1093/imrn/rnz056}, number={11},
journal={International Mathematics Research Notices}, publisher={Oxford University
Press (OUP)}, author={Winkler, Michael}, year={2019}, pages={8106–8152} }'
chicago: 'Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial
Homogeneity in Nutrient Taxis(-Stokes) Systems?” International Mathematics
Research Notices 2021, no. 11 (2019): 8106–52. https://doi.org/10.1093/imrn/rnz056.'
ieee: 'M. Winkler, “Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity
in Nutrient Taxis(-Stokes) Systems?,” International Mathematics Research Notices,
vol. 2021, no. 11, pp. 8106–8152, 2019, doi: 10.1093/imrn/rnz056.'
mla: Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial
Homogeneity in Nutrient Taxis(-Stokes) Systems?” International Mathematics
Research Notices, vol. 2021, no. 11, Oxford University Press (OUP), 2019,
pp. 8106–52, doi:10.1093/imrn/rnz056.
short: M. Winkler, International Mathematics Research Notices 2021 (2019) 8106–8152.
date_created: 2025-12-18T19:35:55Z
date_updated: 2025-12-18T19:59:29Z
doi: 10.1093/imrn/rnz056
intvolume: ' 2021'
issue: '11'
language:
- iso: eng
page: 8106-8152
publication: International Mathematics Research Notices
publication_identifier:
issn:
- 1073-7928
- 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient
Taxis(-Stokes) Systems?
type: journal_article
user_id: '31496'
volume: 2021
year: '2019'
...