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\begin{document}

\title{Introductory Partial Differential Equations}
\author{Final Exam}
\date{May 30, 2013}
\maketitle

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\begin{enumerate}


\item Recall that the solution to the heat equation
\begin{gather*}
  u_t - \Delta u = 0
  \quad \text{in } \mathbb{R}^n \times (0,\infty) \,, \\
  u = g
  \quad \text{on } \mathbb{R}^n \times \{t=0\} 
\end{gather*}
is given by
\[
  u(x,t) = \int_{\mathbb{R}^n} \Phi(x-y,t) \, g(y) \, dy \,,
\]
where, for $t>0$,
\[
  \Phi(z,t) = \frac1{(4\pi t)^{n/2}} \,
              e^{-\tfrac{\lvert z \rvert^2}{4t}} \,.
\]

\begin{enumerate}
\item Show that 
\[
  \sup_{x \in \R^n} \lvert u(x,t) \rvert
  \leq \frac{C}{t^{n/2}}
\]
provided that
\[
  \lVert g \rVert_{L^1}
  = \int_{\R^n} \lvert g(x) \rvert \, \d x < \infty \,.
\]

\item Suppose $w$ solves the Poisson equation 
\[
  -\Delta w = f \qquad \text{in } \R^n 
\]
where $f$ is smooth and compactly supported.  Show that the solution
to the inhomogeneous heat equation
\begin{gather*}
  u_t - \Delta u = f
  \quad \text{in } \mathbb{R}^n \times (0,\infty) \,, \\
  u = g
  \quad \text{on } \mathbb{R}^n \times \{t=0\} 
\end{gather*}
tends to $w$ as $t \to \infty$, i.e., that
\[
  \lim_{t\to\infty} u(x,t) = w(x)
\]
for every $x \in \R^n$.

\end{enumerate}
\hfill(10+10)
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\item Let $U$ be the open unit ball in $\R^n$.  Suppose that $u \in
C_1^2(\overline U_T)$ solves the heat equation
\begin{gather*}
  u_t - \Delta u = 0 \qquad \text{in } U_T \,, \\
  u = g \qquad \text{on } U \times \{t=0\} \,, \tag{H} \\
  u = 0 \qquad \text{on } \partial U \times [0,T] \,,
\end{gather*}
where $U_T = U \times (0,T]$ and $g=0$ on $\partial U$.

\begin{enumerate}
\item Show that a radial solution $u(x,t)\equiv
v(r,t)$ with $r = \lvert x \rvert$ satisfies
\begin{gather*}
  v_t = \frac1{r^{n-1}} \, \frac{\partial}{\partial r} 
        \biggl( r^{n-1} \, \frac{\partial v}{\partial r} \biggr) 
  \qquad \text{in } (0,1) \times (0,T] \,, \\
  v = g \qquad \text{on } (0,1) \times \{t =0 \} \,, \tag{R} \\
  v = 0 \qquad \text{on } \{ r=1 \} \times [0,T] \,.
\end{gather*}

\item Show that if $v \in C_1^2([0,1] \times [0,T])$ solves (R), it is
the unique such solution.

(In particular, show that a boundary condition at $r=0$ is not
required.  Why would you expect this to be so?)

\item When setting up a numerical discretization scheme for (R), a
boundary condition at $r=0$ might be necessary.  What condition could
you suggest?
\end{enumerate}
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\item Consider the linear transport equation
\begin{gather*}
  u_t + b \, u_x = 0 \qquad \text{in } \R \times (0,\infty) \,, \\
  u = g \qquad \text{on } \R \times \{ t = 0 \} \,.
\end{gather*}
We say that $u \in L^\infty(\R \times (0,\infty))$ is an integral
solution of the transport equation provided
\[
  \int_0^\infty \int_\R u \, (v_t + b \, v_x) \, \d x \, \d t
  + \int_\R g(x) \, v(x,0) \, \d x = 0
\]
for all $v \in C_c^\infty(\R \times [0,\infty))$.
\begin{enumerate}
\item Suppose that $g \in C(\R)$ and that $u\in
C^1(\R\times[0,\infty))$ is an integral solution.  Show that $u$ solves
the transport equation in the classical sense.

\item Now suppose that $g(x)$ is bounded and smooth except at some $a
\in \R$ where it has a jump discontinuity.  Show that
$u(x,t)=g(x-bt)$ is an integral solution.
\end{enumerate}
\hfill (10+10)
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\item Find the entropy solution for Burgers' equation
\begin{gather*}
  u_t + u \, u_x = 0 \qquad \text{in } \R \times (0,\infty) \,, \\
  u = g \qquad \text{on } \R \times \{ t = 0 \} \,.
\end{gather*}
with initial data
\[
  g(x) =
  \begin{cases}
    1-x & \text{for } x \in [0,1] \\
    0 & \text{otherwise} 
  \end{cases}
\]
\begin{enumerate}
\item on the interval $t \in (0,1)$;
\item for $t \geq 1$.
\end{enumerate}
\emph{Hint:} Recall that the Rankine--Hugoniot shock condition for a
conservation law of the form $u_t + F(u)_x=0$ states that the solution
at an isolated shock curve parameterized by $(s(t),t)$ satisfies $\dot
s \, [u] = [F(u)]$, the brackets denoting the jump of the enclosed
quantity across the shock.  
\hfill(10+10)

%\newpage\mbox{}\newpage
  
\item Let $u \in C(\R^3,\R^3)$ be a vector field with 
\[
  \lvert u(x) \rvert 
  \leq \frac1{1+\lvert x \rvert^3} \,.
\]
Show that
\[
  \int_{\R^3} \operatorname{div} u \, \d x = 0 \,.
\]
\hfill(10)
%\newpage
\item Let $u(x,t)$ be a smooth solution to the \emph{Korteweg--de
Vries equation}
\[
  u_t - 6 \, u \, u_x + u_{xxx} = 0 
\]
on $\R \times (0,\infty)$ such that for every fixed $t\geq 0$,
$u(x,t)$ and all its derivatives converge to zero as $\lvert x \rvert
\to \infty$.

Show that
\[
  E(t) = \int_\R u(x,t)^2 \, \d x
\]
remains constant in time.
\hfill(10)


\end{enumerate}

\end{document}
