Cauchy–Kovalevskaya for first-order analytic systems

By F1 set $w=u-g(x)$ to obtain $w_t=H(t,x,w,D_{x}w)$ with zero data. Put $c=H(0,0,0,0)$ and $z=w-tc$. Its equation is $z_t=H(t,x,z+tc,D_{x}z)-c=:K(t,x,z,D_{x}z)$, where K is analytic and K(0)=0. Its initial data remain zero. All substitutions stay inside the original analytic neighborhood after a finite shrinking.

F2 gives a unique formal z. Choose a common positive-radius coefficient bound M,r for the finite analytic family K (equivalently bound the absolute series on a smaller polydisc). F4 constructs the convergent positive Goursat majorant for these constants; F3 gives $\mathrm{\mid }[t^q{x}^{\alpha }]z_j\mathrm{\mid }\le [t^q{x}^{\alpha }]U_j$. Choose a positive polyradius s strictly inside the convergence region of U. If $C={\max }_j{\sum }_{\beta }[Z^{\beta }]U_j{s}^{\beta }$, then every coefficient of z is at most $C{s}^{-\beta }$.

F5 makes z an analytic sum on a smaller polydisc and permits its spatial and time derivatives to be taken termwise. Shrink once more so the absolute sums of z and its spatial derivatives are inside the convergence radii of K. Expanding K then gives an absolutely convergent substitution, whose Taylor coefficients agree with the formal substitution in F2. Each coefficient of $z_t-K(t,x,z,D_{x}z)$ is zero by the recursion; hence this function is identically zero there. All coefficients are real, so the restriction is a real analytic solution. Adding tc and g reverses step 1.1 and supplies the prescribed trace.

Any other analytic solution undergoes the same subtraction in step 1.1; its Taylor series solves the identical zero-data formal problem. F2 forces equality of every coefficient with z. On a common smaller neighborhood both analytic functions equal their series and therefore coincide. Reversing the subtraction proves uniqueness of the original germ.