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

Mario Fuest

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 $Ω$, \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 $\overline Ω\times E$ 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 $\frac12$-dimensional Hausdorff measure of $(0, \infty) \setminus E$ vanishes and thus that in particular its Hausdorff dimension is at most $\frac12$. As $\frac12$ 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.

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

Abstract

The chemotaxis-fluid system \begin{align}\tag{}\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 , \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 ) in for some countable union of open intervals with . The present paper investigates further regularity properties of this solution and proves that ( can be chosen such that) the -dimensional Hausdorff measure of vanishes and thus that in particular its Hausdorff dimension is at most . As 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.
Paper Structure (8 sections, 15 theorems, 38 equations)

This paper contains 8 sections, 15 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Main ideas.
  3. Plan of the paper.
  4. Notation.
  5. Global solutions to approximate problems
  6. A new functional inequality
  7. Limits of the solution and the functional y_eps
  8. Estimating the Hausdorff dimension

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded, convex domain and let Then there exist a global weak energy solution $(u, v, w)$ of prob:main in the sense of Definition def:sol below as well as $T_\star \in [0, \infty)$, a set $\mathcal{I} \subseteq \mathbb{N}$ and pairwise disjoint open intervals $I_\iota \subseteq (0, T_\star)$, $\iota \in \mathcal{I}$, such that th has the following pr

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 7 more