On a Conjecture Concerning the Stokes Problem in Nonsmooth Domains
John G. Heywood · Birkhäuser Basel eBooks · 2001
This paper continues the efforts of Xie [], [], [], [], [] and the present author [] to prove an inequality of the form sup0< for solutions of the Stokes equations in an arbitrary three-dimensional domain 1-2, with a constantcindependent of the domain. Here the norms are L2-norms over S2, and A denotes the Stokes operator. The function u is assumed to be solenoidal, and to vanish on the boundary and at spatial infinity. Xie [] proved the inequality modulo one missing point that he left as a conjecture. Recently, in [2], we showed that the desired inequality would also follow from another conjecture that seems to us more approachable. The present paper offers some partial results and observations from our efforts to prove this new conjecture.