solution for a pde on a curve.

Question

Let assume that $ u_t+uu_x=0 $ has $C^1$ solutions in two domains which is disjointed by the curve $x=\phi(t)$. Also assume that $u$ is continuous but $u_x$ has jump discontinuty on the curve. Show that:

$$ \dfrac{d\phi}{dt} = u $$

My method:

I think that if I write the solution in parametric way, by characteristic method, then the rest of the problem is only a computation. But I think there may be another simple aproach.

Answer 1

You cannot use the method of characteristics for a discontinuous solution to a hyperbolic PDE. My approach to this type of problem is as follows:

1. Integrate the PDE across an arbitrary region of $(x, t) $ space. This integral equation should be valid for all regions.
2. Transform integral equation into a new coordinate system $(\xi, \eta) = (x-\phi(t), t-t_0) $ for arbitrary $t_0$ so that the discontinuity is on the curve $\xi=0$
3. Consider the leading order condition given by the integral for the region $\xi\in[-\epsilon, \epsilon] $, $\eta\in [-\delta, \delta] $ as $\epsilon, \delta \rightarrow 0$ if $u$ is discontinuous at $\xi=0$, this is the Rankine-Hugoniot condition.
4. Investigate your discontinuity using the condition, which should give you its speed as a function of the variables of the system.