Global solvability with arbitrary mass for a chemotaxis--fluid model with sensitivity growth between linear and quadratic
Abstract
We study a Patlak--Keller--Segel--Navier--Stokes system on 𝕋×ℝ with density-dependent diffusion and chemotactic sensitivity growing like n1+α, where 0<α<1. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501--543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density--fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that α<1 allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.
Related
- Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension
- A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case
- Boundary layer analysis for the 2D Chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part II. viscous vanishing limit
- Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity
- Global Solvability in Critical Sobolev Spaces for divergence-type equations associated with elliptic systems of vector fields
BibTeX
@misc{wang2026global,
title = {{Global solvability with arbitrary mass for a chemotaxis--fluid model with sensitivity growth between linear and quadratic}},
author = {Lili Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19453},
eprint = {2610.19453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global solvability with arbitrary mass for a chemotaxis--fluid model with sensitivity growth between linear and quadratic
AU - Lili Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19453
UR - https://arxiv.org/abs/2610.19453
AB - We study a Patlak--Keller--Segel--Navier--Stokes system on 𝕋×ℝ with density-dependent diffusion and chemotactic sensitivity growing like n^1+α, where 0<α<1. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501--543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density--fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that α<1 allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.
ER -