A weak-L^p Prodi-Serrin type regularity criterion for a bioconvective flow

ABSTRACT

We determine regularity criteria for weak solutions of a bioconvective flow in a bounded three-dimensional domain. We show that the weak solution $(\mathit{v},\mu )$ is strong on [0, T] if either $\mathit{v}\in L^s((0,T),L^{r,\infty }(\mathrm{\Omega }))$ or ${\Vert \mathit{v}\Vert }_{L^{s,\infty }((0,T),L^{r,\infty }(\mathrm{\Omega }))}$ is bounded from above by a specific constant, where $(3/r)+(2/s)=1$ and $r>3$. No additional condition for $\mu$ is required.

KEYWORDS:

• Navier–Stokes type equations
• bioconvective flow
• strong solutions
• Prodi–Serrin regularity

AMS SUBJECT CLASSIFICATIONS:

• 35Q30
• 76D05
• 76N10

Acknowledgements

This work was finished during the research stay of the third author at the Department of Mathematics, partially funded by the Colóquio de Matemática(CR4430), of the University of Antofagasta.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) [grant number 1120260].