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\begin{document}           


\centerline{{\sc Problem set 6}}
\centerline{Math 218A, Winter 2001}




\bigskip\noindent
{\bf 1.} Use the method of characteristics
to solve the eikonal equation
\[
u_x^2 + u_y^2 = 1
\]
with initial data given parametrically on the curve $y^2 = 2x +1$ by
$x = x_0(s)$, $y = y_0(s)$, $u=u_0(s)$, where
\[
x_0(s) = \frac{\cos s}{1-\cos s},\quad
y_0(s) = \frac{\sin s}{1-\cos s},\quad
u_0(s) = \frac{\cos s}{1-\cos s}.
\]
Show that you get two solutions, depending on the choice of compatible
initial data for $p=u_x$ and $q=u_y$, and that they are given by
\[
u=x,\qquad u = \sqrt{x^2+y^2}-1.
\]

\bigskip\noindent
{\bf 2.} Compute explicitly the Hopf-Lax solution of the Hamilton-Jacobi
equation
\[
\phi_t + \frac12 \phi_x^ 2 = 0,
\]
with the following two sets of initial data:
\beann
&&\phi(x,0) = \left\{
\begin{array}{rl}
x & \mbox{if $x < 0$}, \\
0 & \mbox{if $x \ge 0$},
\end{array}\right.
\\
&&\phi(x,0) = \left\{
\begin{array}{rl}
0 & \mbox{if $x < 0$}, \\
x & \mbox{if $x \ge 0$},
\end{array}\right.
\eeann
Sketch the projections of the characteristics onto
$(x,t)$-space. Compare your result with the corresponding
entropy solution of the inviscid Burgers equation for $u=\phi_x$.


\bigskip\noindent
{\bf 3.} Problems 6 and 7 in \S 3.5, p. 163 of Evans.


\bigskip\noindent
{\bf 4.} Use the method of characteristics and shock-fitting to find an
entropy solution of the initial value problem
\beann
&&u_t + \left(\frac12 u^2\right)_x = 0,\\
&&u(x,1) = \left\{
\begin{array}{rl}
x & \mbox{if $-p < x < q$}, \\
0 & \mbox{otherwise},
\end{array}\right.
\eeann
where $p, q \ge 0$. At what rate does $\|u\|_\infty$ decay
as $t\to +\infty$?

\smallskip
{\sc Remark.} This solution is called an ``N-wave''. It is the time-asymptotic
state of any compactly supported solution of the inviscid Burger's equation. The
front and rear shocks in an N-wave can be heard in sonic booms.
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