EMAIL FROM MARK PINSKY QUOTING
JEFFREY HUMPHREYS

ON THE IMPORTANCE OF OUR WORK

ON THE LINEARIZED BOLTZMANN
EQUATION

(boldface added in text of emails)

Date:
Mon, 16 Jun 2003 18:11:05 -0500 (CDT)

From: Mark Pinsky <pinsky@math.northwestern.edu>

To: rsellis@math.umass.edu

Subject: Re: your mail

Dear Richard,

**. . . **Quite
coincidentally, last week there was a PDE conference at Northwestern, at which
time I as approached by Jeff Humpherys of Ohio State,
who is a careful reader of our papers from 1974-75, applying the ideas in all
sorts of new directions. I share with you, for your possible interest, his
message below. **It appears that the field is still alive and well and that
our work has played an important part in the further progress of the subject.**
(The "reprint" refers to my JMPA paper from 1976, a sequel to our
joint work, where I cleaned up the asymptotics w.r.t. Navier Stokes). It also
occurs to me that Bruce Turkington may have some peripheral interest in these
matters. **. . . **

Mark P;

On Mon, 16 Jun 2003, Jeff Humpherys wrote:

Date:
Mon, 16 Jun 2003 17:04:28 -0400 (EDT)

From:
Jeff Humpherys <jeffh@math.ohio-state.edu>

To:
pinsky@math.nwu.edu

Dear
Mark,

It
was nice to meet you last Thursday at the conference. I appreciate your giving me the reprint. I was unfamiliar with its existence and the
corresponding/related work of Kurtz and also of Papanicolaou
and Kohler that you provide in the bib.

Below
is the BibTeX ref for Kawashima's paper from
1985. **It borrows heavily from your
paper with Ellis, and is of substantial importance in the study of viscous
conservation laws.** I've recently
expanded on their work by applying your ideas (and theirs) to conservation laws
of higher order and have found necessary and sufficient conditions for the admissibility
of symmetrizable viscous-dispersive and higher order
fronts and pulses. By admissibility in
this context, I mean that the essential spectrum coming from linearization
about a constant solution is well behaved, i.e., there is no essential spectrum
in the right half-plane. This is a
necessary condition for stability of asymptotically constant traveling waves,
i.e., fronts, pulses, etc.

The
key to their results, which is consistent with those that you did with Navier-Stokes, is that these systems are genuinely coupled,
meaning, the eigenvectors of the flux, A:=Df, are not
in the kernel of the viscosity term B.
In fact, Kawashima's staggering result says that for symmetrizable
systems, admissibility is equivalent (!) to genuine coupling, and is also
equivalent to the existence of a skew-symmetric K such that

B
= \Pi_A(B) + [K,A], (*)

where
Pi_A(B) is positive definite (all of this notation is
consistent with both your paper and Kawashima's). Note that B is only non-negative definite,
and so uniform bounds on estimates using B are essentially impossible. It is the existence of this K, and hence, \Pi_A, that allows one to do energy estimates. For example, in my thesis, I proved linear
stability for symmetrizable (degenerately) viscous
conservation laws. Others have used
Kawashima's work to prove numerous other important results. For example there's work by Matsumura,
Nishihara, Nishita, etc. in the study of viscous
shocks. Also, there are results in both conservation laws and kinetic theory by
Yong, T-P Liu, Zumbrun, and others which use this as
well.

What
I've done is extended Kawashima to higher order conservation laws. So now one
can have a viscous B, dispersive C, etc., and for symmetrizable
systems, again, genuine coupling is equivalent to admissibility, and also to
the existence of this skew-symmetric K such that a higher-order version of (*)
holds.

**Anyway,
I can go on and on, but I wanted you to know that your work with Ellis is very
important.**

Best
regards,

Jeffrey
Humpherys

@article
{MR86k:35107,

AUTHOR = {Shizuta,
Yasushi and Kawashima, Shuichi},

TITLE = {Systems of equations of
hyperbolic-parabolic type with

applications to the discrete {B}oltzmann equation},

JOURNAL = {Hokkaido Math. J.},

FJOURNAL = {Hokkaido Mathematical Journal},

VOLUME = {14},

YEAR = {1985},

NUMBER = {2},

PAGES = {249--275},

ISSN = {0385-4035},

CODEN = {HMAJDN},

MRCLASS = {35M05 (35B40 76P05 82A40)},

MRNUMBER = {86k:35107},

MRREVIEWER
= {Carlo Cercignani},

}

--

Mark A. Pinsky

Department of Mathematics

Northwestern University

Evanston, IL 60208-2730

Tel:847-491-5519

Fax: 847-491-8906