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

Abstract

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 $\Omega$, \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 $\overline{\Omega }\times E$ for some countable union of open intervals $E$ with $|(0,\infty )\setminus E|=0$. 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 $(0,\infty )\setminus E$ 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.

arXiv comment13 pages

References in corpus (1)

• Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system