On L-q estimates of planar flows up to the boundary

Abstrakt

We show apriori L-q gradient estimates for a sufficiently smooth planar flow driven by generalized Stokes system of equations. The estimates are obtained up to the boundary of a container where the fluid is contained.

We allow power growth p - 1, p is an element of (1, +infinity) of the extra stress tensor for large values of shear rate but we exclude degeneracy or singularity for small shear rate. We also allow arbitrary q is an element of [p, +infinity).

The technique does not provide L-q theory since we need to assume that the forces are very smooth.

Klíčová slova

• Generalized Stokes problem
• differentiability
• regularity up to the boundary