Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 30E / b / Solution

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.