TITLE: How does research look in higher order ($>2$) PDE?
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: 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!