Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 30E (Partial Differential Equations)

The first-order Cauchy-Kovalevskaya theorem says that an analytic PDE system solved for the derivative normal to an analytic noncharacteristic initial hypersurface, with analytic initial data, has a unique local analytic solution near each point of that hypersurface. The general theorem likewise applies to higher-order equations solved for the highest normal derivative with analytic data for the lower normal derivatives.

Here the characteristic vector is $(x,y,a)$ and the normal to $z=0$ is $(0,0,1)$. Its normal component is $a$, so the theorem applies precisely when $a\ne 0$. The method of characteristics gives $\dot{x}=x$, $\dot{y}=y$, $\dot{z}=a$, $\dot{u}=u$. Starting from $(\xi ,\eta ,0)$ at parameter $s=0$ gives $x=\xi e^s$, $y=\eta e^s$, $z=as$, $u=e^{s}f(\xi ,\eta )$. Eliminating $s=z/a$ yieldsThis is locally defined wherever the arguments remain in the analytic domain of $f$, and verifies both PDE and initial data.

As $a\to 0$, the normal component vanishes and the characteristic curves become tangent to the initial plane. The crossing parameter $s=z/a$ and slopes $dx/dz=x/a$, $dy/dz=y/a$ become singular. At $a=0$, arbitrary analytic data are not admissible: already on the plane they must obey $x{f}_x+y{f}_y=f$. Even compatible data do not determine derivatives normal to the plane. This is failure of the noncharacteristic condition, not a failure of the analytic theorem.

ChooseDirect differentiation gives $u_{ny}=v_{nx}$ and $v_{ny}=-u_{nx}$. Both functions are smooth and $2\pi$-periodic in $x$, with initial data $f_n(x)=n^{-1}\cos nx$, $v_n(x,0)=0$. Their squared norms arefor every fixed $y\ne 0$. Thus continuous dependence on the initial data fails in this $L^2$ topology, even on smooth analytic data. The Cauchy problem for a partial differential equation is ill-posed in the sense of Hadamard well-posedness; existence of analytic solutions from the preceding theorem does not supply stability for an elliptic Cauchy problem.