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\begin{document}

\title{Partial Differential Equations}
\author{Final Exam}
\date{December 19, 2002}
\maketitle

\textbf{Get 50 out of 70 points for a 100\% score.}

\begin{enumerate}

\item
\begin{enumerate}
\item  Show that the Fourier transform for $f\colon \R \to \R$,
\[
  f(x) = e^{-t |x|} \,,
\]
is given by
\[
  \hat f (\xi) = \sqrt{\frac2\pi} \, \frac{t}{\xi^2+t^2} \,.
\]

\item Use the result from (a) to solve the 1-dimensional differential
equation
\[
  u_{xx} - u = \delta \,,
\]
where $\delta$ is the Dirac measure centered at $x=0$.

\end{enumerate}
\hfill(5+5)


\item Assume that $u \in C^3 (\R^n \times [0,\infty))$ solves the heat
equation
\[
  u_t = \Delta u \,.
\]
Prove that $v \equiv \lvert Du \rvert^2 + u_t^2$ is a
\emph{subsolution} of the heat equation, i.e.
\[
  v_t \leq \Delta v \,.
\]
\hfill(10)

\item Find an \emph{a priori} estimate which can be used to prove that
if $u \in \h1(\R)$, then $u$ is continuous.

Hint: Use the Fundamental Theorem of Calculus to estimate
$|u(x)-u(y)|$.
\hfill(10)

\item Show that
\[
  E(t) = \frac12 \int_\R u_t^2 \, dx
       + \frac12 \int_\R u_x^2 \, dx
\]
is a constant in time for smooth solutions $u \colon
\R\times[0,\infty) \to \R$ \emph{with compact support in space} of the
wave equation
\[
  u_{tt} - u_{xx} = 0 \,.
\]
\hfill(10)

\item A function $G_L$ is called the Green's Function of a linear
partial differential operator $L$ if
\[
  L \, G_L = \delta \,,
\]
where $\delta$ is the Dirac measure centered at zero.  Show, given two
partial linear differential operators $A$ and $B$ with $A+B=1$, that
\[
  G_{AB} = G_A + G_B \,.
\]
Note: You may assume without further discussion that expressions like
$A \delta$ and $B \delta$ can be made sense of.
\hfill(10)

\item Consider the initial value problem for Burger's equation 
\begin{gather*}
  u_t + u \, u_x = 0 \,,  \\
  u(0) = u^\ini \,,
\end{gather*}
where $u \colon \R \times [0, \infty) \to \R$, and we write $u(t)$ as
shorthand for $u(\,\cdot\,,t)$. 

\begin{enumerate}
\item Show that, provided $u \in C^1(\R \times [0,\infty))$ and
$u^\ini \in L^2(\R)$,
\[
  \norm{u(t)}{\l2} = \norm{u^\ini}{\l2} \,. \tag{*}
\]

\item Show that, provided $u \in C^1(\R \times [0,\infty))$,
\[
  \int_0^\infty \int_\R
    \bigl(
      \phi_t \, u + \tfrac12 \, \phi_x \, u^2
    \bigr) \, dx \, dt
  = -\int_\R \phi(x,0) \, u^\ini(x) \, dx
  \tag{**}
\]
for every test function $\phi \in C^1(\R \times [0,\infty))$ with
compact support.

\item It is known that for $0 \leq t \leq 2$ the following function is
a so-called \emph{entropy solution} of Burger's equation:
\[
  u(x,t) = \begin{cases}
    0 & \text{for } x \leq 0 \\
    \frac{x}{t} & \text{for } 0 < x < t \\
    1 & \text{for } t \leq x < 1 + \frac{t}2 \\
    0 & \text{for } x \geq 1 + \frac{t}2
  \end{cases}
\]
Show, by direct calculation, that $u$ does \emph{not} satisfy (*).

\item Show, again by direct calculation, that $u$ as defined in (c)
satisfies (**).

\end{enumerate}
\hfill(5+5+5+5)

\end{enumerate}

\end{document}
