Solving the Shock Wave Problem for $u_t + uu_x = 0$

[evinda]

Hello! (Wave)

I want to solve the equation $u_t+uu_x=0$ with the initial condition $u(x,0)=1$ for $x \leq 0$, $1-x$ for $0 \leq x \leq 1$ and $0$ for $x \geq 1$. I want to solve it for all $t \geq 0$, allowing for a shock wave. I also want to find exactly where the shock is and show that it satisfies the entropy condition.

I have tried the following.

We get that $u(x(t),t)=c$.

The characteristic line that passes through the points $(x,t)$ and $(x_0,0)$ has slope

$\frac{x-x_0}{t-0}=\frac{dx}{dt}=u(x,t)=u(x_0,0)=\left\{\begin{matrix}
1, & \text{ if } x_0 \leq 0,\\
1-x_0, & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$$\Rightarrow x-x_0=\left\{\begin{matrix}
t, & \text{ if } x_0 \leq 0,\\
t(1-x_0), & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$

$\Rightarrow u(x,t)=\left\{\begin{matrix}
1, & \text{ if } x \leq t,\\
\frac{1-x}{1-t}, & \text{ if } t \leq x \leq 1,\\
0, & \text{ if } x \geq 1.
\end{matrix}\right.$Do we have a shock at the time when $u$ is not continuous? But how can we find such a time?

 

[I like Serena]

Hello! (Wave)

I want to solve the equation $u_t+uu_x=0$ with the initial condition $u(x,0)=1$ for $x \leq 0$, $1-x$ for $0 \leq x \leq 1$ and $0$ for $x \geq 1$. I want to solve it for all $t \geq 0$, allowing for a shock wave. I also want to find exactly where the shock is and show that it satisfies the entropy condition.

I have tried the following.

We get that $u(x(t),t)=c$.

The characteristic line that passes through the points $(x,t)$ and $(x_0,0)$ has slope

$\frac{x-x_0}{t-0}=\frac{dx}{dt}=u(x,t)=u(x_0,0)=\left\{\begin{matrix}
1, & \text{ if } x_0 \leq 0,\\
1-x_0, & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$$\Rightarrow x-x_0=\left\{\begin{matrix}
t, & \text{ if } x_0 \leq 0,\\
t(1-x_0), & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$

$\Rightarrow u(x,t)=\left\{\begin{matrix}
1, & \text{ if } x \leq t,\\
\frac{1-x}{1-t}, & \text{ if } t \leq x \leq 1,\\
0, & \text{ if } x \geq 1.
\end{matrix}\right.$Do we have a shock at the time when $u$ is not continuous? But how can we find such a time?

Hey evinda!

What will happen to $u$ when $t$ reaches $1$? (Wondering)

Btw, your solution doesn't cover $t>1$ does it?
The second condition breaks down, and the first and third condition will overlap. (Worried)

 

[evinda]

The shock wave happens when $t=1$.

Which is the entropy condition? (Thinking)

 

[I like Serena]

The shock wave happens when $t=1$.

Which is the entropy condition?

I don't know what an entropy condition is. (Crying)

Do you have a definition? (Wondering)

 

[evinda]

I don't know what an entropy condition is. (Crying)

Do you have a definition? (Wondering)

I found the definition at page $12$: https://web.stanford.edu/class/math220a/handouts/conservation.pdf.

In our case, ee have $f'(u)=u$.
At $t=1$ we have $u^-=1$ and $u^+=0$, and so we have $u^- > u^+$, which means that the entropy condition is satisfied.

Right? (Thinking)

 

[I like Serena]

I found the definition at page $12$: https://web.stanford.edu/class/math220a/handouts/conservation.pdf.

In our case, ee have $f'(u)=u$.
At $t=1$ we have $u^-=1$ and $u^+=0$, and so we have $u^- > u^+$, which means that the entropy condition is satisfied.

Right?

Yes, I believe so. (Nod)

 

[evinda]

Yes, I believe so. (Nod)

Nice, thank you! (Smile)