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example from AMS literature

To: w3cmatherb@w3.org

Subject: example from AMS literature

From: Ron Whitney <RFW@math.ams.org>

Date: Mon, 01 Jul 1996 23:21:42 0400 (EDT)

From RFW@math.ams.org Mon Jul 1 23: 21:50 1996

MailSystemVersion: <MultiNetMM(369)+TOPSLIB(158)+PMDF(5.0)@MATH.AMS.ORG>

MessageId: <836277702.203453.RFW@MATH.AMS.ORG>
Both at Neil's behest and because I'd like to accumulate a set of test
cases anyway, I started looking for some notational examples in AMS
literature. As is almost always the case when I start searching for
these things, the first article I ran across supplies some matters for
thought and is perhaps worth some discussion.
Below is a rekeyboarding of the very beginning of the article in TeX.
(So if there are TeX errors, I'm the source.) The definitions are
simply the standard TeX definitions for the div and curl operators;
\lim and \max are similarly defined. If a display file (say,
PostScript) would also help, let me know. I'm hoping that all on this
list are somewhat conversant in TeX. The % signs are simply aids to
readability (% is the TeX comment character).
Below the example are some notes. Exercises: (1) how will this be
encoded in HTMLMath, and (2) how will the semantics be handled?
Comments on the difficult or odd parts are all that is required.
************************************************************************
\def\div{\operatorname{\rm div}}
\def\curl{\operatorname{\rm curl}}
The Euler equations for an inviscid incompressible 2D fluid flow are
given by
%
$$\aligned & D\nu/Dt = \nabla p, &\qquad& x\in R^2, t>0 \\
& \div\nu = 0, &\qquad& \nu(x,0) = \nu_0(x)
\endaligned$$
%
where $\nu = {}^t(\nu_1,\nu_2)$ is the fluid velocity, $p$ is the
scalar pressure, $D\nu/Dt = \partial\nu/\partial t +
(\nu\cdot\nabla)\nu$, and $\nu_0$ is an initial incompressible
velocity field, i.e.~$\div\nu_0=0$.
In this paper, we study the detailed limiting behavior of approximate
solution sequences for 2D Euler with vortex sheet initial data. A
sequence of smooth velocity fields $\nu^\epsilon(x,t)$ is an {\it
approximate solution sequence} for 2D Euler provided that the $\nu$
is incompressible, i.e.~\div \nu=0$, and satisfies the following
properties:
(1) The velocity fields $\nu^\epsilon$ have uniformly bounded local
kinetic energy, i.e.
%
$$\max_{0\leq t\leq T}\int_{x\leq R} \nu^\epsilon(x,t)^2\,dx\leq C$$
%
for any $R,T>0$.
(2) The corresponding vorticity, $\omega^\epsilon=\curl\nu^\epsilon$,
is uniformly bounded in $L^1$, i.e.
%
$$ \max_{0\leq t\leq T}\int\omega^\epsilon(x,t)\,dx\leq C$$
%
for any $T>0$.
(3) the vortex field $\nu^\epsilon$ is weakly consistent with 2D
Euler, i.e.~for all smooth test functions, $\phi\in
C^\infty(R^2\times(0,\infty))$ with $\div \phi=0$,
%
$$ \lim_{\epsilon\rightarrow0}\int\int\phi_t\cdot\nu^\epsilon +
\nabla\phi : \nu^\epsilon\otimes\nu^\epsilon\,dx\,dt=0.$$
%
Here $\nu\otimes\nu = (v_i v_j)$, $\nabla\phi =
(\partial\phi_i/\partial x_j)$, and $A:B$ denotes the matrix product
$\sum_{i.j}a_{ij}b_{ij}$. We remark in passing ...
************************************************************************
Notes:
1. "R" is used both to mean the reals and as a bound variable in the second
display.
2. \nu is a fluid velocity; \nu_1 and \nu_2 are its spatial components,
whereas \nu_0 is its time slice at t=0.
3. A prescript is used (just under the first display) to indicate
functional dependence.
4. The / is used for differentiation (so, of course, D\nu/Dt binds
as (D\nu)/(Dt) and not D(\nu/D)t ).
5. Superscripts have a variety of meanings (no surprise): cartesian product,
sequence index, power. L^1 and C^\infty show functional dependence
upon their superscripts and these might be considered of the same ilk
as \nu^\epsilon (where \epsilon is an index).
6. Parens show functional dependence, pairing, grouping
(i.e. precedence), interval specification, and matrices.
7. An ad hoc operator (:) is defined for matrix product.