Chemotaxis-Stokes system with p-Laplacian slow diffusion: global boundedness across the full p>2 range
Abstract
The global boundedness problem for three-dimensional chemotaxis-fluid systems with nonlinear diffusion presents substantial analytical challenges, particularly in the near-linear diffusion regime. Indeed, the porous-medium and p-Laplacian diffusion operators approach the classical Laplacian as msearrow1 and psearrow2, and the extra a priori control afforded by nonlinear structures becomes increasingly difficult to exploit. This difficulty is reflected in the porous-medium setting, where establishing boundedness throughout m>1 required a sequence of substantial developments spanning more than a decade (2010-2024). For the p-Laplacian case, despite significant progress, the problem remained open in the near-linear range 2<p≤ p*(≈2.01), due to the saturation of classical Moser-type iterations as psearrow2. In this paper, we establish global bounded strong solutions for a three-dimensional chemotaxis-Stokes system with p-Laplacian slow diffusion and signal consumption. Our approach departs from the conventional Lq-iteration framework and develops a weighted energy method that directly exploits the coercive structure of the nonlinear diffusion operator. A weighted Rellich identity yields density estimates. To control the resulting boundary contribution on general smooth domains, we first use tangential differentiation along the boundary to lower its order, and then apply a trace-embedding estimate. Combined with a high-gradient/low-gradient decomposition, these estimates allow us to control the mixed chemotactic terms and close the coupled estimates for the density, signal, and fluid. The resulting estimates hold throughout 2<p≤ 3, thereby resolving the previously open range 2<p≤ p*. Together with existing results for p>p*, our work establishes boundedness throughout the entire slow-diffusion range p>2.
Related
- Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity
- Global Boundedness and Finite-Time Blow-up for a Quasilinear Chemotaxis--May--Nowak System
- Improved Global Boundedness Criteria via Refined Bootstrap Estimates for a Quasilinear Chemotaxis-May-Nowak System
- Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension
- Boundary layer analysis for the 2D Chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part II. viscous vanishing limit
BibTeX
@misc{jin2026chemotaxisstokes,
title = {{Chemotaxis-Stokes system with $p$-Laplacian slow diffusion: global boundedness across the full $p>2$ range}},
author = {Chunhua Jin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01847},
eprint = {2611.01847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01847},
}
RIS
TY - UNPB
TI - Chemotaxis-Stokes system with p-Laplacian slow diffusion: global boundedness across the full p>2 range
AU - Chunhua Jin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01847
UR - https://arxiv.org/abs/2611.01847
AB - The global boundedness problem for three-dimensional chemotaxis-fluid systems with nonlinear diffusion presents substantial analytical challenges, particularly in the near-linear diffusion regime. Indeed, the porous-medium and p-Laplacian diffusion operators approach the classical Laplacian as msearrow1 and psearrow2, and the extra a priori control afforded by nonlinear structures becomes increasingly difficult to exploit. This difficulty is reflected in the porous-medium setting, where establishing boundedness throughout m>1 required a sequence of substantial developments spanning more than a decade (2010-2024). For the p-Laplacian case, despite significant progress, the problem remained open in the near-linear range 2<p≤ p^*(≈2.01), due to the saturation of classical Moser-type iterations as psearrow2. In this paper, we establish global bounded strong solutions for a three-dimensional chemotaxis-Stokes system with p-Laplacian slow diffusion and signal consumption. Our approach departs from the conventional L^q-iteration framework and develops a weighted energy method that directly exploits the coercive structure of the nonlinear diffusion operator. A weighted Rellich identity yields density estimates. To control the resulting boundary contribution on general smooth domains, we first use tangential differentiation along the boundary to lower its order, and then apply a trace-embedding estimate. Combined with a high-gradient/low-gradient decomposition, these estimates allow us to control the mixed chemotactic terms and close the coupled estimates for the density, signal, and fluid. The resulting estimates hold throughout 2<p≤ 3, thereby resolving the previously open range 2<p≤ p^*. Together with existing results for p>p^*, our work establishes boundedness throughout the entire slow-diffusion range p>2.
ER -