Abstract
We prove existence of a solution to the divergence equation satisfying a new Bogovski-type estimate for the difference quotients. This enables us to give an alternative proof of the interior regularity of the solution to the p-Stokes problem, completely avoiding the pressure. Moreover, as a key preliminary result we prove boundedness of Calderón–Zygmund operators with standard kernels in weighted Lebesgue and Orlicz spaces over a general domain.
Highlights
We are concerned with the question of interior regularity of the weak solution of the steady Stokes approximation for flows of shear thinning fluids
We show that the solution of the divergence equation obtained via the Bogovski formula satisfies an additional estimate for the difference quotient
The proof of this is based on estimates of singular Calderón–Zygmund operators generated by standard kernels in arbitrary domains Ω ⊂ Rn
Summary
We are concerned with the question of interior regularity of the weak solution of the steady Stokes approximation for flows of shear thinning fluids. U = 0 on ∂Ω, where Ω ⊂ Rn, n ≥ 2, is a bounded domain. Un) denotes the unknown velocity vector field, and π the unknown scalar pressure, while the external body force f = ∇u ), the symmetric part of the velocity gradient ∇u. The relevant example we have in mind is
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