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Perhaps this is a question inappropriate for this site, but I would like to know if research exists in PDEs of order higher than 2, and if so, what it looks like. As an addendum, I'd also like to ask - why is the order 2 case such a fundamental object of study?

I took a number of graduate courses in PDE, and while it is a huge field, and I barely scratched the surface, the most general equations we'd see would have terms like:

$$ F(x,u,\nabla u, D^2u) = 0 $$ with no third-order terms. Of course, proving things about this usually boiled down to some convexity argument , so it makes sense to see $D^2u$ here, but the only 3rd order (or higher) PDE I have studied is the Korteweg DeVries Equation: $$ u_t + u_{xxx} + 6uu_x = 0 $$ mostly as an exercise in Fourier Analysis.

All of this comes down to the question - is there active research in PDEs of order higher than 2? I know that order 2 PDE come from nature (Newton's laws, Generator of Diffusion processes), but what about say order 3? Or order 7? Are these equations of interest? And if so, are there any" "laws of nature" that might motivate their formulation?

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    $\begingroup$ The question in the title seems to be different from the one in the last paragraph. But as far as examples are concerned, there's the biharmonic equation, many other water wave equations besides KdV, and so on. You'll find plenty of examples in Zwillinger's Handbook of Differential Equations, Section 40. $\endgroup$ Jun 24, 2021 at 14:13
  • $\begingroup$ @HansLundmark thanks - this was sort of a stream of consciousness writeup. I'll change the title. $\endgroup$ Jun 24, 2021 at 14:14
  • $\begingroup$ Thank you @GiuseppeNegro I really like this approach. If you could explain why exactly rotational invariance, or perhaps the presence of various symmetries, is a desirable or fundamental property of "physical laws", like you might to a 5-year old, I would appreciate it! It has been a while since I have taken a physics course. Also, if you could outline the proof that no third-order or higher operators can be rotationally invariant (I have seen the proof that the laplacian/its powers are the unique linear, second-order operator that is rotationally invariant), I would also appreciate that. $\endgroup$ Jun 24, 2021 at 14:57
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    $\begingroup$ @rubikscube09: Your question is quite vague as currently stated, and inviting long rambling discussions in comments is not a very good use of this site. A better approach might be to do some research into topics suggested in the comments and to use that research to ask a more focussed question. $\endgroup$
    – Lee Mosher
    Jun 24, 2021 at 17:27
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    $\begingroup$ @LeeMosher: I have removed my "rambling" comments. Don't close this question and don't make it CW, please. I will write a coherent answer sooner or later. $\endgroup$ Jun 25, 2021 at 11:40

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I'd argue that the interest in low-order equations is that they tend to arise from physical problems, and for some folks that's attractive. As far as PDEs of more general orders go, I'd suggest starting with vol 2 of Hörmander's "The Analysis of Linear Partial Differential Operators." Good question!

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