# What conditions constrain PDEs which are studied in practice?

In general, it is naive to ignore applications when studying or looking for motivations for theoretical objects in PDEs. Nearly all applications of PDEs are in physical sciences, engineering, materials science, image processing, computer vision, etc. These are the motivations for studying particular types of PDEs, and without these applications, there would be almost zero mathematical interest in many of the PDEs we study. For instance, why do we spend so much time studying parabolic and elliptic equations, instead of focusing effort on bizarre fourth order equations like $$u^π_{xxxx}=u^2_ye^{u_z}$$? (hint: there are physical applications of elliptic and parabolic equations). We study an extremely small sliver of all possible PDEs, and without a mind towards applications, there is no reason to study these PDEs instead of others.

I was wondering, are there any ways to (partially) characterize this sliver?

When originally asking this question, I used the word "interesting" as shorthand for the PDEs I wanted to ask about. This was unhelpful since "interesting" is an inherently subjective notion. Below is a hopefully less-subjective definition.

Let $$P$$ be a property of PDEs; I'll characterize what I'm looking for in $$P$$ (there's probably multiple viable candidates, hence the soft-question tag). The idea is: $$P$$ should be well-defined, and "mathematically natural" in the sense of having a short mathematical characterization, but also broad enough to include the "small sliver" of PDEs which are actually studied. [*]

Asking for $$P$$ to correspond exactly to this sliver would obviously be too much, because "actually studied" is not a mathematical property, is not even a binary property [**], and changes over time. On the other hand, I think asking for necessary (but not sufficient) conditions might be reasonable. Here's why:

In practice, new PDEs are often derived from old ones via operations such as: placing constraints on certain variables / parameters, or the relationships between them; taking limits as they approach particular values; changing co-ordinates; substituting one expression with a related but simpler one which may give rise to similar behaviour. I can imagine applying such constructions finitely (countably?) many times to obtain a sort of closure operator on classes of PDE, and find it conceivable that if $$P$$ is closed in this sense and also large enough, it could include our sliver.

To recap: "actually used by scientists, engineers etc." is sufficient (but not necessary) for $$P$$ to hold. $$P$$ should also be closed under the sorts of operations used in practice to derive new PDEs from old ones. Taking the closure under such operations will make $$P$$ not only broad but also, I hope, simple to characterize. At the same time, I would still expect it to exclude pathological examples like $$u^π_{xxxx}=u^2_ye^{u_z}$$.

[*] The analogy with Reverse Mathematics, mentioned in comments below, is: there are weak subtheories of second-order arithmetic which are broad enough to prove the mathematical results actually used in science and engineering; they turn out to be "natural" not only in having short characterizations, but also because many alternative candidate theories turn out to be equivalent.

[**] I wrote this question with the interesting number paradox in mind; the template above, with integers instead of PDEs, would just give all of them.

So: are there any natural candidates for $$P$$? And if so, how narrow can those candidates be before they start ruling out things with potential applications?

• I think that the quote text does a pretty good job of explaining which "natural" conditions make a PDE "interesting". A PDE is "interesting" precisely when it models a physical phenomenon. That is, there is nothing intrinsic to mathematics which makes PDE "interesting". – Xander Henderson Aug 8 '19 at 11:57
• That's not a mathematically natural property, because it's aiming to be both necessary and sufficient. I'm looking for properties / conditions which are mathematically natural, and do not aim to be sufficient. – Robin Saunders Aug 8 '19 at 12:01
• I know that it isn't a mathematically natural property. That was precisely my point. "Interesting" PDEs are interesting because they model the world in which we live. This is something external to mathematics. – Xander Henderson Aug 8 '19 at 12:43
• Even the "necessary" condition that an equation has to make sense (possibly with the consideration of boundary or initial conditions) in order to be interesting is not purely a mathematical property; an observer is involved in making "sense" of a problem. – hardmath Aug 8 '19 at 12:52
• Surely a PDE can be mathematically interesting without having any physical (or other “external”) applications! To give just one example, consider Novikov's equation $(u-u_{xx})_t = (3 u_x u_{xx} - 4 u u_x + u u_{xxx})u$ (mathscinet.ams.org/mathscinet-getitem?mr=2430566), which has no applications whatsoever as far as I know, but is studied quite intensely by PDE people because of its intriguing behaviour when it comes to blowup of solutions, (non)uniqueness of weak solutions, complete integrability, etc. – Hans Lundmark Aug 8 '19 at 13:52