A weak-\(L^p\) Prodi-Serrin type regularity criterion for a bioconvective flow. (English) Zbl 1421.35249
Summary: We determine regularity criteria for weak solutions of a bioconvective flow in a bounded three-dimensional domain. We show that the weak solution \((\nu,\mu)\) is strong on \([0,T]\) if either \(\nu \in L^s ((0,T), L^{r,\infty} (\Omega))\) or \(\Vert \nu\Vert_{L^{s,\infty}((0,T), L^{r,\infty}(\Omega))}\) is bounded from above by a specific constant, where \((3/r)+(2/s)=1\) and \(r>3\). No additional condition for \(\mu\) is required.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76Z10 | Biopropulsion in water and in air |
| 35D35 | Strong solutions to PDEs |
Keywords:
Navier-Stokes type equations; bioconvective flow; strong solutions; Prodi-Serrin regularityReferences:
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