Are there standard approaches, to solving a system of nonlinear PDE?

Question

If we have a system of PDE's, where each PDE is different i.e. for

$u:U\subset \Bbb R^2\to\Bbb R^3$, $u(x,y)=(a(x,y),b(x,y),c(x,y))$, which needs to satisfy

$ \left\{ \begin{array}{ll} u_x\cdot u_y=0\\ |u_x|^2-|u_y|^2=0 \\ H(u)=c\end{array} \right.$

Where $H(u)$ is the mean curvature of $u$, and $c$ is a constant.

The reason I ask, is that it seems different to a system $u_t=u_{xx}$ (for example) where $u$ is a vector field. Here the PDE is the same in each case.

Is there a specific theoretical approach to take?

Thanks for any help.

Answer 1

it seems different to a system $u_t=u_{xx}$ (for example)

It is different. The linear heat equation on $\mathbb R^n$ is one of the PDE for which we can write down a general solution. There are other such PDE... but you can count them on the fingers of one hand.

Are there standard approaches, to solving a system of nonlinear PDE?

Yes. The standard approach is: you don't solve a system of nonlinear PDE. You decide on the function space in which it should be posed. You consider the appropriateness of initial/boundary conditions and forcing term, if applicable. You carefully approach the problems of existence and uniqueness. And of regularity of solutions. Their asymptotic behavior. Comparison principles (like the maximum principle). And so on. But you don't solve them. Because you can't.

The system you wrote down says that $u$ is a conformal CMC immersion. So, the study of that PDE is the study of CMC surfaces in $\mathbb R^3$, a subject with which books have been filled. The special form of the first two equations (conformality) opens a connection with complex analysis, from which the subject benefits (holomorphic Hopf differential; Kenmotsu's representation formula). So, this system is a lot better understood that what you'd get by, say, changing the first equation to $u_x\cdot u_y = 1$ and keeping the rest the same.

Suggestion: if you are interested in the PDE perspective on constant mean curvature surfaces, begin by reading the lecture notes Constant Mean Curvature Surfaces: Harmonic Maps and Integrable Systems by Frederic Hélein. They are more readable than most books concerning nonlinear PDE. There is also a short book Surfaces with Constant Mean Curvature by Kenmotsu, but it's not really organized as a book from which one could learn the subject.