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- Apr 13, 2013

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Hello!!!

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 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?

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