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{ \large \textbf{\textsc{Math 5588 -- Homework 11 (Due Thursday April 20)}}}
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\begin{enumerate}
\item A weak solution of Burger's equation 
\[u_t + uu_x =0 \ \ \text{ for } x \in \R, \ t>0\]
has the form
\[u(x,t) = \begin{cases}
\frac{x}{t+1},& \text{for } x < s(t)\\
0,& \text{for } x > s(t),
\end{cases}\]
where $s(t)$ is a shock curve starting at $s(0)=1$.  Find a formula for $s(t)$. [Hint: Use the Rankine-Hugoniot condition to find an ODE that $s(t)$ satisfies.]
\item Find the solution of Riemann's problem, which is Burger's equation
\[u_t + uu_x = 0\]
with initial condition $u(x,0)=f(x)$ where
\[f(x) = \begin{cases}
u_l,&\text{if } x<0\\
u_r,&\text{if } x>0,
\end{cases}\]
where $u_l$ and $u_r$ are constants describing the initial states to the left and right of zero, respectively. You can assume $u_l\neq u_r$. [Hint: Depending on whether $u_l < u_r$ or $u_l > u_r$ you will have either a rarefaction wave or a shock wave. Give the solution in both cases.]
\item Recall the Legendre transform of a function $H(p)$ is given by
\[H^*(q) = \max_{p \in \R} \{ pq - H(p)\}.\]
Find the Legendre transform of $H(p) = \frac{1}{2}p^2$ and show directly that $H^{**}=H$. [Here, $H^{**}=(H^*)^*$.]
\item Show that the Legendre transform of the function $H(p) = \frac{1}{r}|p|^r$ for $1 < r < \infty$ is
\[H^*(q) = \frac{1}{s}|q|^s\]
where $\frac{1}{s} +\frac{1}{r} = 1$.
\end{enumerate}
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