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I'm trying to solve the following transport equation

$$\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi}{\partial x}=0$$

subject to the initial condition $$\phi(x,0)=f(x)=\left\{ \begin{array}{l l} 1,\quad x<0 \\ 0, \quad x\geq0 \end{array} \right. $$

This is my solution:

$\frac{dx}{dt}=\phi \Rightarrow x=\phi t+C$. At $t=0$ let $x=s$. Therefore $x=\phi t+s$. So along a characteristic we have $s=x-\phi t$.

For $s<0$, $\phi=1$ so $s=x-t$.

For $s\geq0$, $\phi=0$ so $s=x$.

$\frac{d\phi}{dt}=0 \Rightarrow \phi(x,t)=f(x)$ and given $\phi=f(s)$ at $x=s,t=0$ we find that $$u(x,t)=f(s)=\left\{\begin{array}{l l} 1,\quad s<0 \\ 0, \quad s\geq0 \end{array} \right.$$

$$\Rightarrow \phi(x,t)=\left\{\begin{array}{l l} 1,\quad x-t<0 \\ 0, \quad x\geq0 \end{array} \right.$$

Apparently the answer is $$ \phi(x,t)=\left\{\begin{array}{l l} 1,\quad t>2x \\ 0, \quad t<2x \end{array} \right.$$

I'm not sure where I went wrong. Can someone please tell me where my working is wrong?

1 Answers1

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As you found, on the region $\{(x,t): 0\le x<t\}$ the characteristics with $s=x$ and $s=x-t$ intersect, indicating the existence of a shock. Thus the solution is $$ ϕ(x,t)=\begin{cases}1& x<s(t),\\0& x\ge s(t)\end{cases} $$ where the shock wave front $s(t)$ moves at the average speed of the characteristics intersecting there, in this case $s'(t)=\frac{f(x)+f(x-t)}2=\frac12$, thus $s(t)=\frac t2$.

For other instructive examples of shock waves in Burger's equation see the answers in

Lutz Lehmann
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