Longer or alternate versions of papers in my desk drawer
For some papers it can be hard to decide how much detail to include for more-or-less standard arguments, and how far to stray from the main thread by adding tangential, albeit interesting, results. Without the need to have a paper accepted for publication, however, one can write at greater length. I have included a few instances, here.
Contains some proofs and speculations not really appropriate for a published version, being too much detail or too conjectural.
Also includes a proof of a 2001 result of Xiaoming Wang's, using the technology developed in the paper. I did not include
this proof in the submitted version because it is no shorter than the original argument, and differs significantly from it only in that
it works out the result for a time-varying boundary layer width. It also brings out the importance of a flat boundary to
the argument; whether or not a curved boundary can be handled, I do not know.
In summer 2019 I ran across a 2010 paper by J. Simon on the dangers of identifying
the classical space H of incompressible fluid mechanics, divergence-free vector fields
in L^2 tangential to the boundary, with its own dual space. I decided to write up a set of
notes making aspects of this issue, "Simon's trap" if you will, as concrete as possible.
This led me on a huge detour into a type of 3D stream function for elements of H which vanish
on the boundary, which I had found useful, in a more limited version, in my paper
"Vanishing viscosity and the accumulation of vorticity on the boundary." That then took
over the focus of my efforts and led me on a long journey into some geometric literature
where, it turns out, such things, though not there called stream functions, were known
in any dimension (though mostly with smooth boundaries). This ultimately resulted
in my paper, "On stream functions for divergence-free vector fields," published in 2020,
in which I finally realized how to achieve what I wanted for Lipschitz
domains using no geometry. But that was only after I had figured out how to do it the
hard way. The hard way is more constructive, however, and has the potential to
provide more information. So I have included here some longer versions of the paper
including the constructive, somewhat geometric approach, a description of the relation
to the geometric results in the literature, along with a few digressions into
tangentially related issues.
Two other versions containing only parts of the extended version are here:
Constructive version with no
digressions, Version with digressions only
A talk on this paper from December 2015. Of particular note is a simple restatement
of the result in Lagrangian coordinates. (See the corollary on page 15.)
This paper supersedes The vortex patches of Serfati, extending the result to higher dimensions, simplifying
some of the proofs, and correcting a few small errors as well. The older paper includes a number of 2D examples, however.
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