Last week, when Michael asked for a list of fundamental theorems in different branches of mathematics, Juan de Mairena suggested the Lax Equivalence Theorem as a candidate. Today on ArXiv I spotted a paper that makes the rather dramatic claim that the theorem is “wrong” — not that it is wrong in the strict mathematical sense, but that its conditions are not realistic for real-world problems. I’m not in a position to evaluate the claim (I never even heard of the result until Juan’s comment), but I thought it was interesting to see a paper on the subject so soon after we discussed it.

Does that mean he has to give back his prize?

I read the paper… and I can’t understand his point.

The main objections in 1 and 2… well, the theorem is valid only for linear pde! [see P.D. Lax R. D. Richtmyer, Stability of difference equations, Comm. Pure and Applied Math, IX (1956) 267-293 or the following doctoral thesis http://www.math.lsu.edu/grad/zhuang.pdf ] However, he ‘proved’ it for nonlinear operators… perhaps is his own paper the wrong one?

From “3. Is completeness an appropriate requirement ?” and the last sections, another objection seems to be the completitude of the space… which is fulfilled in the Lax-Richt. paper in section 2 (or 3, I haven’t got it here now). Of course, there are several good references about those things and generalizations in excellent journals like Numerische Mat., or Math. Comp.

Also, sect. 4 is… the “boundedness” condition 5.18 is the uniform boundedness of the operators (the continuity of the operators, in functional analysis, and not the topological notion of a bounded set…) The principle of unif. bound. doesn’t need the compacity of the set of operators!

Also, sect. 4 isâ€¦ the â€œboundednessâ€ condition 5.18 is the [remove uniform] boundedness of the operators…

The uniform bound. is obtained from the UBP, sorry for the mistake and for my poor english!

An interesting thing about the applicability for real-world problems, beside the linearity of the equations, the only two conditions are:

-well posed problems: small changes in the data gives small changes on the solutions [any physical problem assume it, due to the inaccuracy of measurements]

-consistency: the numerical escheme converges to the equation when the step size goes to zero [the physical way to solve differential equations: just change derivatives by incremental quotients!]