I've made an approximation to HTML markup (of the math) for the example I posted yesterday. In the ensuing portion of this message, I've enclosed some notes and comments, followed by alternation between TeX and HTML markup of the passages. Questions about semantics are of course much more difficult to answer, but I'd be interested in any comments people have on how they think the semantics of this example should be treated (or rather how our "pre-semantics" will set things up). And please do let me know if the HTML markup needs adjustment. -Ron ************************************************************************ Notes and questions: 1. I haven't made an effort to substitute character names as they exist in TR 9573. By and large, I've substituted SGML-like entities for TeX characters. Identifiers such as \div in TeX come out \div in our current proposal, although it's my understanding that "\div\nu" changes to "\div ν" since ν is letter-like. True? 2. The differentials "Dx", etc. have been changed to "ⅅx", etc. Should "\partial\nu" be changed to "&ppartial;ν" since the \partial symbol itself might be reserved for meaning "boundary" and &ppartial; provides some mnemonic consistency? [[BTW, later in this article we have integrals with differential "d|\omega^\epsilon|(y)"]] I take it that the form "differential &over; differential" will be interpreted as a derivative by software which renders the expression tree semantically. Bruce spoke of visual renderers having the freedom to display &over; with an inline /. Does the freedom also exist in reverse? 3. I've left the alignment of the first display for a later iteration. The <sep> and <?space> markup only indicate a need to have column separators and some indication of "separation" or distinction between equation and condition. 4. I added SGML markup to "<mo>(ν\cdot\nabla)</mo>" since the expression within the parens is an operator. The parens are not properly part of the operator, though. How should this have been handled? 5. How is "\div ν" interpreted? It appears to be two identifiers juxtaposed and is therefore implicitly a product of the two. I've inserted &FunctionApplication; where no parens appeared after the operator. 6. Have we decided that we can use the _ and ^ characters for the lower and upper limits of sums and integrals? 7. I'm guessing that we won't be able to simply enter | uncharacterized as left or right (when the | occurs in a matching pair), so I've used &leftvert; and &rightvert; for the pair. 8. BTW, in using the convenient screen display form for <math> elements, I've segregated these elements so that the text of the paragraph after </math> starts on a new line. I take it this adds significant space in an SGML document. 9. The \max and \lim identifiers are juxtaposed to the left of integral signs. Is the relation interpreted as multiplication? Should I have placed a &FunctionApplication; entity between the two? 10. None of the bracketing operators should stretch in these passages. Should they have been marked as stretchy="false" somehow? 11. I added some <mo> markup to the ad hoc : operator of the last display. Left and right precedences are undefined. I take it that the same locution has to be carried through to all other occurrences of : (with the same meaning). ** TeX ******************************************************************* The Euler equations for an inviscid incompressible 2-D 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.~$\div\nu_0=0$. ** HTML ****************************************************************** The Euler equations for an inviscid incompressible 2-D fluid flow are given by <math> <alignment_structure> <mrow> <sep> ⅅν/ⅅt = -∇p, <sep><?space><sep> x&element; R^2, t>0 </mrow> <mrow> <sep> \div&FunctionApplication;ν = 0, <sep><?space><sep> ν(x,0) = ν_0(x) </mrow> </alignment_structure> </math> where <math mode="inline">ν= (ν_1,ν_2)^^^t</math> is the fluid velocity, <math mode="inline">p</math> is the scalar pressure, <math mode="inline">ⅅν/ⅅt = &ppartial;ν/&ppartial;t + <mo>(ν\cdot\nabla)</mo>ν</math>, and <math mode="inline">ν_0</math> is an initial incompressible velocity field, i.e. <math mode="inline">\div&FunctionApplication;ν_0=0</math>. ** TeX ******************************************************************* In this paper, we study the detailed limiting behavior of approximate solution sequences for 2-D Euler with vortex sheet initial data. A sequence of smooth velocity fields $\nu^\epsilon(x,t)$ is an {\it approximate solution sequence} for 2-D Euler provided that the $\nu$ is incompressible, i.e.~$\div \nu=0$, and satisfies the following properties: ** HTML ****************************************************************** In this paper, we study the detailed limiting behavior of approximate solution sequences for 2-D Euler with vortex sheet initial data. A sequence of smooth velocity fields <math mode="inline">ν^ε(x,t)</math> is an <em>approximate solution sequence</em> for 2-D Euler provided that the <math mode="inline">ν</math> is incompressible, i.e. <math mode="inline">\div&FunctionApplication;ν=0</math>, and satisfies the following properties: ** TeX ******************************************************************* (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 $R,T>0$. ** HTML ****************************************************************** (1) The velocity fields <math mode="inline">ν^ε</math> have uniformly bounded local kinetic energy, i.e. <math>\max_{0≤ t≤ T}∫_{&leftvert;x&rightvert;≤ R} &leftvert;ν^ε(x,t)&rightvert;^2ⅆx≤ C</math> for any <math mode="inline">R,T>0</math>. ** TeX ******************************************************************* (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 $T>0$. ** HTML ****************************************************************** (2) The corresponding vorticity, <math mode="inline">ω^ε= \curl&FunctionApplication;ν^ε</math>, is uniformly bounded in <math mode="inline">L^1</math>, i.e. <math> \max_{0≤ t≤ T} ∫&leftvert;ω^ε(x,t)&rightvert;ⅆx≤ C </math> for any <math mode="inline">T>0</math>. ** TeX ******************************************************************* (3) the vortex field $\nu^\epsilon$ is weakly consistent with 2-D 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 $\nu\otimes\nu = (\nu_i \nu_j)$, $\nabla\phi = (\partial\phi_i/\partial x_j)$, and $A:B$ denotes the matrix product $\sum_{i,j}a_{ij}b_{ij}$. We remark in passing ... ** HTML ****************************************************************** (3) the vortex field <math mode="inline">ν^ε</math> is weakly consistent with 2-D Euler, i.e. for all smooth test functions, <math mode="inline">φ&element; C^&infinity;(R^2×(0,&infinity;))</math> with <math mode="inline">\div&FunctionApplication;φ=0</math>, <math> \lim_{ε→0} ∫∫φ_t·ν^ε + ∇φ<mo infix="true" leftprec=? rightprec=?>:</mo> ν^ε&circletimes;ν^ε ⅆx ⅆt=0. </math> Here <math mode="inline">ν&circletimes;ν = (\nu_i \nu_j)</math>, <math mode="inline">∇φ = (&ppartial;φ_i/&ppartial; x_j)</math>, and <math mode="inline">A<mo infix="true" leftprec=? rightprec=?>:</mo>B</math> denotes the matrix product <math mode="inline">∑_{i,j}a_{ij}b_{ij}</math>. We remark in passing ...Received on Tuesday, 2 July 1996 22:44:01 UTC
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