# Ngô Quốc Anh

## March 31, 2010

### Wave equations: Shock Formation

Filed under: PDEs — Ngô Quốc Anh @ 0:01

So far once we have discontinuous initial data, we might have shocks. This also occurs even the given initial data is smooth. Let us still consider the following problem

$\displaystyle u_t+c(u)u_x=0,\quad x\in \mathbb R, t>0$

subject to the following initial condition

$\displaystyle u(x,0)=u_0(x), \quad x\in \mathbb R$

where $c(u)>0$, $c'(u)>0$ and $u_0 \in C^1$.

From a discussion of this entry we consider the case when $u_0$ is non-increasing as we need singularities (because $c'>0$). Moreover, the solution can be given implicitly as

$\displaystyle\left\{ \begin{gathered} u(x,t) = {u_0}(\xi ), \hfill \\ x - \xi = c({u_0}(\xi ))t. \hfill \\ \end{gathered} \right.$

Since $u_0$ is non-increasing, $u'_0(x)<0$ on $\mathbb R$. We now consider the characteristics which are straight lines, issuing from two points $\xi_1$ and $\xi_2$ on the $x$ axis with $\xi_1 < \xi_2$ have speeds $c(u_0(\xi_1))$ and $c(u_0(\xi_2))$ respectively. Because $u_0$ is decreasing and $c$ is increasing, it follows that

$\displaystyle c(u_0(\xi_1))>c(u_0(\xi_2))$.

In other words, the characteristic emanating from $\xi_1$ is faster than the one emanating from $\xi_2$.

Therefore the characteristics cross so a smooth solution cannot exist for all $t>0$.

We are now interested in calculating the breaking time. To this purpose, we calculate $u_x$ along a characteristic which has equation

$x-\xi=c(u_0(\xi))t$.

Let $g(t)=u_x(x(t),t)$ denote the gradient of $u$ along the characteristic $x=x(t)$ given as above. Then

$\displaystyle \frac{dg}{dt}=u_{tx}+c(u)u_{xx}$.

By differentiating the PDE with respect to $x$ we also have

$\displaystyle u_{tx}+c(u)u_{xx}+c'(u)u_x^2=0$.

Comparing gives

$\displaystyle \frac{dg}{dt}=-c(u)g^2$

along the characteristic. Solving this ODE gives us

$\displaystyle g=\frac{g(0)}{1+g(0)c'(u_0(\xi))t}$

where $g(0)$ is the initial gradient at $t=0$. Thus

$\displaystyle {u_x} = \frac{{{{u'}_0}(\xi )}}{{1 + {{u'}_0}(\xi )c'({u_0}(\xi ))t}}$.

The fact that $u'_0$ and $c'$ have different sign implies that $u_0$ will blow up at a finite time along the characteristic. What we need to do is to examine $u_0$ along all characteristics to find such $\xi$ so that $u_x$ first blows up.

A simple calculation show by denoting $F(\xi)=c(u_0(\xi))$ that the time of the first breaking is

$\displaystyle t_b=-\frac{1}{F'(\xi_b)}$

where $\xi_b$ is such that $|F'(\xi)|$ is maximum. An in-depth observation shows that if the initial data is not monotone, breaking will first occur on the characteristic $\xi=\xi_b$ for which $F'(\xi)<0$ and $|F'(\xi)|$ is maximum.

Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.3.