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