Existence and boundedness of the classical global solution to 2D chemotaxis-fluid system with double chemical signals

This article concerns the chemotaxis-fluid system with double chemical signals, $$\displaylines{ n_t + u \cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla c)+\xi \nabla \cdot (n\nabla v) + f(n), \quad x \in \Omega, t > 0, \cr c_t + u \cdot \nabla c=\Delta c-nc, \quad x \in \Omega,\; t > 0, \cr v_t + u \cdot \nabla v= \Delta v-v+n, \quad x \in \Omega,\; t > 0, \cr u_t =\Delta u+\nabla P+n\nabla \phi, \quad x \in \Omega, \;t > 0, \cr \nabla \cdot u = 0, \quad x \in \Omega,\; t > 0, }$$ in a smooth bounded domain $\Omega\subset \mathbb{R}^2$ with no-flux/no-flux/no-flux/no-slip boundary conditions. Here $f(n)$ is a given sub-logistic source function satisfying $f(s)\geq \zeta s$ for small $s\geq0$, where $\zeta\in \mathbb{R}$, and the growth condition $$ \liminf_{s \to \infty} \big\{-f(s) \frac{\ln s}{s^2}\big\} = \mu\in (0, \infty]. $$ We obtain the existence and uniform-in-time boundedness of classical global solutions to the corresponding initial-boundary value problem. We assume that the initial data and the physical coefficients satisfy $$ \left(2 + \chi^2 + \xi^2 + 4\|c_0\|_{L^\infty(\Omega)}^2 + 4 C_{\mathrm{p}}^2\|c_0\|_{L^\infty(\Omega)}^2 \|\nabla \phi\|_{L^\infty(\Omega)}^2 -\mu\right)^+ < \frac{1}{2 C_{\mathrm{GN}}^4 M}, $$ where \(M\) is a constant, $$ M := \| n_0 \|_{L^1(\Omega)} + |\Omega| \inf_{\eta > 0} \frac{\sup \{ f(s) + \eta s : s > 0 \}}{\eta}, $$ $C_{\mathrm{GN}} > 0$ is the Gagliardo-Nirenberg constant, and $C_\mathrm{p}$ the Poincare constant. This result reveals that the sub-logistic source plays a crucial role in preventing blow-up phenomena within the chemotaxis-fluid system with double chemical signals. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/10/abstr.html