Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems

The chemotaxis-fluid system \begin{align}\tag{$\star$}\label{prob:star} \begin{cases} n_t + u \cdot \nabla n = Δn - \nabla \cdot (n \nabla c), \ c_t + u \cdot \nabla c = Δc - nc, \ u_t + (u \cdot \nabla) u = Δu + \nabla P + n \nabla Φ, \quad \nabla \cdot u = 0, \end{cases} \end{align} models aerobic bacteria interacting with a fluid via transportation and buoyancy. When posed on a three-dimensional, smoothly bounded, convex domain $\mathrm{\Omega }$, \eqref{prob:star} complemented with suitable initial and boundary conditions is known to admit a global `weak energy solution’, which recently has been shown to be smooth (after a redefinition on a set of measure $0$) in $\bar{\mathrm{\Omega }}\times E$ for some countable union of open intervals $E$ with $\mathrm{\mid }(0,\mathrm{\infty })\setminus E\mathrm{\mid }=0$. The present paper investigates further regularity properties of this solution and proves that ($E$ can be chosen such that) the $\frac{1}{2}$-dimensional Hausdorff measure of $(0,\mathrm{\infty })\setminus E$ vanishes and thus that in particular its Hausdorff dimension is at most $\frac{1}{2}$. As $\frac{1}{2}$ has been the best known upper estimate for the Hausdorff dimension of the temporal singular set for the unperturbed Navier–Stokes equations for quite some time, this result is the best one can hope for \eqref{prob:star} without significant progress in the regularity theory of (homogeneous) Navier–Stokes equations.