Leray-type partial regularity of weak solutions for a chemotaxis-fluid system with porous medium diffusion

In this paper, we consider the coupled chemotaxis-fluid system (0.1) ⎧ { { { ⎨ { { { ⎩ 𝑛 𝑡 + 𝑢 · 𝛻 𝑛 = Δ ⁢ 𝑛 𝑚 − 𝛻 · ( 𝑛 𝛻 𝑐 ) , 𝑥 ∈ Ω , 𝑡 > 0 , 𝑐 𝑡 + 𝑢 · 𝛻 𝑐 = Δ ⁢ 𝑐 − 𝑛 ⁢ 𝑐 , 𝑥 ∈ Ω , 𝑡 > 0 , 𝑢 𝑡 + ( 𝑢 · 𝛻 ) ⁢ 𝑢 = Δ ⁢ 𝑢 + 𝛻 𝑃 + 𝑛 𝛻 𝜙 , 𝑥 ∈ Ω , 𝑡 > 0 , 𝛻 · 𝑢 = 0 , 𝑥 ∈ Ω , 𝑡 > 0 in a smoothly bounded domain Ω ⊂ ℝ 3 , subject to no-the flux boundary conditions, where m > 1. A recent result [1] concerning the regularity property of solutions to (1.0) in the case 𝑚 = 1 reveals that the temporal singular sets of the global weak solutions constructed in [2] have Lebesgue measure zero. Motivated by this, and building upon the existence theory for global weak solutions to (1.0) established in [3, 4], the present work shows that an analogous regularity property remains valid for any m > 1. More precisely, on ̅ ̅ ̅ ̅ ̅ Ω × 𝐸 , the solution component n is continuous, while c and u are smooth, where E is a countable union of open intervals satisfying | ( 0 , ∞ ) ∖ 𝐸 | = 0 .