\b[Compatible <Cauchy data>] for a <wave map> are $\phi_0:\mathbb R^n\to S^2$ and $\phi_1(x)\in T_{\phi_0(x)}S^2$: thus $|\phi_0|=1$ and $\phi_0\cdot\phi_1=0$. Smooth localized <wave map Cauchy data> equal a constant $p\in S^2$ with zero velocity outside a compact set. It is $\phi_0-p$, rather than the <sphere>-valued map itself, that has <compact support>. The geometric constraints propagate under the <wave map> equation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 10 / 4 / 1 / Solution

 Solution (source code)