Global Existence and Partial Regularity for the Three-Dimensional Chemotaxis--Navier--Stokes System in a Half-Space
Abstract
We establish endpoint partial regularity for suitable weak solutions to the three-dimensional chemotaxis--Navier--Stokes system with constant sensitivity and linear consumption in a half-space under no-slip and no-flux boundary conditions. The full space--time singular set, including its boundary part, has vanishing one-dimensional parabolic Hausdorff measure. This follows from a Caffarelli--Kohn--Nirenberg type criterion at the natural r−1 dissipation scaling, valid at interior and boundary points without additional pressure smallness. The key estimate combines weighted mass control with the localized entropy--energy inequality to close the iteration without a separate nlog n quantity and accounts for the mass term generated by rescaling. Boundary-compatible pressure and caloric decompositions yield local Hölder regularity. We also prove a lower-order criterion requiring smallness only of the velocity, signal gradient, and pressure oscillation, with a threshold depending on an arbitrary finite density mass bound. We construct global suitable weak solutions under natural entropy and energy assumptions without smallness restrictions on the initial data. The potential is time-independent with bounded gradient; boundedness of the potential itself is assumed only for the large-time results. If its gradient is also Hölder continuous, a solution can be selected to become bounded and classical throughout the closed half-space after a finite waiting time. For the same solution, we obtain decay and Neumann Gaussian asymptotics for both scalar components without initial spatial moment assumptions. The profile coefficients are the conserved bacterial mass and the limiting signal mass, respectively, and the latter is strictly positive whenever the initial signal is nonzero.
Related
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BibTeX
@misc{chen2026global,
title = {{Global Existence and Partial Regularity for the Three-Dimensional Chemotaxis--Navier--Stokes System in a Half-Space}},
author = {Yubo Chen and Jianbo Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13613},
eprint = {2610.13613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global Existence and Partial Regularity for the Three-Dimensional Chemotaxis--Navier--Stokes System in a Half-Space
AU - Yubo Chen
AU - Jianbo Yu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13613
UR - https://arxiv.org/abs/2610.13613
AB - We establish endpoint partial regularity for suitable weak solutions to the three-dimensional chemotaxis--Navier--Stokes system with constant sensitivity and linear consumption in a half-space under no-slip and no-flux boundary conditions. The full space--time singular set, including its boundary part, has vanishing one-dimensional parabolic Hausdorff measure. This follows from a Caffarelli--Kohn--Nirenberg type criterion at the natural r^−1 dissipation scaling, valid at interior and boundary points without additional pressure smallness. The key estimate combines weighted mass control with the localized entropy--energy inequality to close the iteration without a separate nlog n quantity and accounts for the mass term generated by rescaling. Boundary-compatible pressure and caloric decompositions yield local Hölder regularity. We also prove a lower-order criterion requiring smallness only of the velocity, signal gradient, and pressure oscillation, with a threshold depending on an arbitrary finite density mass bound. We construct global suitable weak solutions under natural entropy and energy assumptions without smallness restrictions on the initial data. The potential is time-independent with bounded gradient; boundedness of the potential itself is assumed only for the large-time results. If its gradient is also Hölder continuous, a solution can be selected to become bounded and classical throughout the closed half-space after a finite waiting time. For the same solution, we obtain decay and Neumann Gaussian asymptotics for both scalar components without initial spatial moment assumptions. The profile coefficients are the conserved bacterial mass and the limiting signal mass, respectively, and the latter is strictly positive whenever the initial signal is nonzero.
ER -