Relation between different forms of Cauchy-Kovalevskaya Theorem

Question

Consider the following form of Cauchy-Kovalevskaya (CK) theorem for a system of PDE:

For a given system of PDE with $p$ unknown functions $u^{(1)}, \dots, u^{(p)}$ and variables $x, y_1, \dots, y_n$ of the form $$ u^{(1)}_x=G_1(u^{(1)}, \dots, u^{(p)}, u^{(1)}_{y_1}, \dots, u^{(p)}_{y_1}\dots,u^{(1)}_{y_n}, \dots, u^{(p)}_{y_n})\\ \vdots\\ u^{(p)}_x=G_p(u^{(1)}, \dots, u^{(p)}, u^{(1)}_{y_1}, \dots, u^{(p)}_{y_1}\dots,u^{(1)}_{y_n}, \dots, u^{(p)}_{y_n}) $$ subject to initial conditions $$u^{(1)}(0, \mathbf{y})=g_1(\mathbf{y})\\ \vdots\\ u^{(p)}(0, \mathbf{y})=g_p(\mathbf{y})$$ suppose that $g_1, \dots, g_p$ are holomorphic in a neighborhood of the point $\mathbf{y}^0$ and $G_1, \dots, G_p$ are holomorphic in a neighborhood of the value of their arguments evaluated at $(0, \mathbf{y}^0)$. Then there exists a neighborhood of $(0, \mathbf{y}^0)$ where the system above has a holomorphic solution satisfying the initial conditions.

Taking this theorem for granted, I want to use it to prove another version of CK theorem for a single PDE of order $m$ in the variables $x, \mathbf{y}$ of the form $$D^m_xu=F(x, \mathbf{y}, u, D_x^kD_y^{\alpha}u \text{ with } k<m \text{ and } k+|\alpha|\le m)$$ satisfying the initial conditions $$D^i_xu(0, \mathbf{y})=f_i(\mathbf{y})$$ for $0\le i<m$.

I am familiar with the idea of associating a system of first order equations to a single $m$th order PDE with initial values, however I suspect that in that process the functions $G_i$ in the system will also depend on the variables $x, \mathbf{y}$ just as $F$ does, so the above form of CK theorem is not immediately applicable. Does anyone know a way around this issue?

As another question, does the statement of CK above remain valid if $G_i$ depend on the variables $x, \mathbf{y}$? (For instance, when the equations have non-constant coefficients). [Edit: The answer to this question is affirmative. See my 2nd comment below.]

Any thoughts on this are welcome.

Answer 1

(Converted and expanded from a comment.)

You can artificially add to your list of unknowns the coordinate functions $x$ and $\mathbf{y}$. They solve first order PDEs of the form $$ \partial_x x = 1 $$ and $$ \partial_x \mathbf{y} = 0.$$

So starting with a system $\mathbf{v}$ that solves $$ \partial_x \mathbf{v} = \mathbf{g}(x,\mathbf{y}, \mathbf{v}, \partial_\mathbf{y} \mathbf{v}) $$ the enlarged system $\mathbf{V} = (x,\mathbf{y}, \mathbf{v})$ now solves a system of the form $$ \partial_x \mathbf{V} = \mathbf{G}(\mathbf{V}, \partial_{\mathbf{y}} \mathbf{V}) $$ as required.

Answer 2

I do not think you can apply the first theorem to the second statement you want to prove, because there is an essential difficulty: The first theorem only applies to holomorphic functions. The classical Cauchy-Kovalevskaya Theorem also need to assume the functions are analytic. Otherwise I think there are counter-examples.

For a correct statement of the theorem and the proof, check the corresponding proof in Evans or Tao's book nonlinear dispersive equations, Chapter 1, where this is given as a homework execrise with some hints. There are also plenty of proofs online. The proof (I did at least) is very technical and you need a careful control of the terms involved. The confusion you have should evaporate once you boil down to the base case.