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

verfasst von

Mario Fuest

Abstract

The chemotaxis-fluid system (Formula presented) models aerobic bacteria interacting with a fluid via transportation and buoyancy. When posed on a three-dimensional, smoothly bounded, convex domain Ω, the above equation 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 Ω × E for some countable union of open intervals E with |(0, ∞) \ E| = 0. The present paper investigates further regularity properties of this solution and proves that (E can be chosen such that) the ¹₂ -dimensional Hausdorff measure of (0, ∞) \ E vanishes and thus that in particular its Hausdorff dimension is at most ¹₂ . As ¹₂ 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 the above system without significant progress in the regularity theory of (homogeneous) Navier–Stokes equations.

Organisationseinheit(en)

Institut für Angewandte Mathematik

Typ

Artikel

Journal

Proceedings of the American Mathematical Society

Band

153

Seiten

2053-2066

Anzahl der Seiten

14

ISSN

0002-9939

Publikationsdatum

20.02.2025

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik, Angewandte Mathematik

Elektronische Version(en)

https://doi.org/10.1090/proc/17148 (Zugang: Geschlossen)
https://doi.org/10.48550/arXiv.2402.16582 (Zugang: Offen)