Characteristic solution of a scalar conservation law
= Characteristic solution of a scalar conservation law
{title2=$u(t,X_t(\xi))=u_0(\xi)$}
Before <characteristic crossing>, the <scalar conservation law> $u_t+f(u)_x=0$ has solution $u(t,X_t(\xi))=u_0(\xi)$ with $X_t(\xi)=\xi+t f'(u_0(\xi))$. When this map is a <diffeomorphism>, the solution is $u_0\circ X_t^{-1}$. Its spatial derivative is $u_0'(\xi)/(1+t f''(u_0(\xi))u_0'(\xi))$.