Time regularity of flows of non-Newtonian fluids with critical power-law growth

TL;DR

This paper investigates the time regularity of solutions to three-dimensional non-Newtonian fluid flows with critical power-law growth, extending regularity and uniqueness results to a broader range of the power index p.

Contribution

It establishes time regularity of solutions for non-Newtonian fluids with p ≥ 11/5, filling a gap in the existing theory and extending uniqueness results.

Findings

Proves regularity of solutions for p ≥ 11/5

Extends uniqueness results to the entire range p ≥ 11/5

Fills the gap between previous results for p ≥ 12/5 and the full energy range

Abstract

We deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index $p\ge 11/5$ we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power--law indexes $p\ge 11/5$. Such a result was available for $p\ge 12/5$ and therefore the paper fills the gap and extends the uniqueness result to the whole range of $p$'s for which the energy equality holds.