In 1985, V. Scheffer discussed partial regularity results for what he called
solutions to the
Navier–Stokes inequality. These maps essentially satisfy the
incompressibility condition as well as the local and global energy inequalities and the
pressure equation which may be derived formally from the Navier–Stokes system
of equations, but they are not required to satisfy the Navier–Stokes system
itself.
We extend this notion to a system considered by Fang-Hua Lin and
Chun Liu in the mid 1990s related to models of the flow of nematic liquid
crystals, which include the Navier–Stokes system when the director field
is
taken to be zero. In addition to an extended Navier–Stokes system, the Lin–Liu model
includes a further parabolic system which implies an a priori maximum principle for
which they use to establish partial regularity (specifically,
) of
solutions.
For the analogous
inequality one loses this maximum principle,
but here we nonetheless establish the partial regularity result
, so that in particular the
putative singular set
has space- time Lebesgue measure zero. Under an additional assumption on
for any fixed value of a
certain parameter
— which
for
reduces precisely
to the boundedness of
used by Lin and Liu — we obtain the same partial regularity
()
as do Lin and Liu. In particular, we recover the partial regularity result
() of
Caffarelli–Kohn–Nirenberg [1982] for suitable weak solutions of the Navier–Stokes system, and
we verify Scheffer’s assertion that the same holds for solutions of the weaker
inequality as well.
We remark that the proofs of partial regularity both here and in the work of Lin
and Liu largely follow the proof in Caffarelli–Kohn–Nirenberg, which in turn used
many ideas from an earlier work of Scheffer [1975].
