Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915234908733440 |
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| author | Fuest, Mario |
| author_facet | Fuest, Mario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems Fuest, Mario Analysis of PDEs 35B65 (primary), 35K55, 35Q35, 35Q92, 92C17 (secondary) 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. |
| title | Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems |
| topic | Analysis of PDEs 35B65 (primary), 35K55, 35Q35, 35Q92, 92C17 (secondary) |
| url | https://arxiv.org/abs/2402.16582 |