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\begin{document}

\title{Partial Differential Equations}
\author{Homework 5}
\date{due December 7, 2009}
\maketitle

\begin{enumerate}

\item Evans, p.\ 426, Problem 6

\item Assume $W \subset \R^n$ is open, $u \colon \R^n \times [0,T] \to
\R^n$ is a smooth time-dependent vector field with flow $\phi_t$,
i.e.\ $\partial_t \phi_t(a) = u(\phi_t(a),t)$, and $W_t = \phi_t(W)$.

Use Liouville's theorem to prove that 
\begin{enumerate}
\item $\displaystyle \frac{d}{dt} \int_{W_t} \rho \, f \, dx
= \int_{W_t} \rho \, (\partial_t + u \cdot \nabla) f \, dx$, where
$\partial_t \rho + \nabla \cdot (\rho u) = 0$, and
\item $\displaystyle \frac{d}{dt} \int_{W_t} f \, dx =
\int_{W_t} \partial_t f \, dx + \int_{\partial W_t} f \, \nu \cdot u
\, dS$.
      
\end{enumerate}

\item Recall the construction of the function spaces for solutions of
the Navier--Stokes equations, where for $U \in \R^n$ open, bounded,
with $C^2$ boundary,
\[
  \mathcal V = \{ u \in C_c^\infty(U, \R^n) \colon \nabla \cdot u = 0 \} \,, 
\]
$H$ is the closure of $\mathcal V$ in $L^2$, and $V$ is the closure of
$\mathcal V$ in $H^1$.  Recall the continuous trace operator $\gamma
\colon E(U) \to H^{-1/2}(\partial U)$ which satisfies $\gamma(u)=\nu
\cdot u$ if $u \in C^\infty(\overline U)$, where
\[
  E(U) = \{ u \in L^2(U,\R^n) \colon \nabla \cdot u \in L^2(U,\R) \} 
\]
with the divergence understood in the sense of weak derivatives. 

Prove that
\[
  H = \{ u \in L^2(U,\R^n) \colon \nabla \cdot u = 0 \text{ and }
  \gamma(u)=0 \} \,.
\]

\item In the notation of the previous question, let $P$ denote the
$L^2$-orthogonal projector onto $H$ and define the \emph{Stokes
operator} $A \colon \D(A) \to H$ by $A = - P\Delta$ with domain $\D(A)
= H^2(U,\R^n) \cap V$.

Recall from class that if $f \in H$, then $A^{-1}f=u$ is the unique
solution in $\D(A)$ of the Stokes equations.  (This was proved by an
application of the a Lax--Milgram theorem followed by an elliptic
regularity argument.)

\begin{enumerate}

\item Show that 
\[
  \langle Au, v \rangle_{L^2(U,\R^n)}
  = \langle u,v \rangle_{H^1(U,\R^n)}
\]
for all $u,v \in \D(A)$, thus conclude that $A$ is a symmetric
operator.  

\emph{Note:} First assume $u,v \in \mathcal V$, then argue by density.
Note also that
\[
  \langle u,v \rangle_{H^1(U,\R^n)}
  = \int_U \nabla u : \nabla v \, dx \,.
\]

\item Show that $A$ is self-adjoint.

\emph{Note:} By part (a), $A$ is symmetric, so it remains to be shown
that $\D(A)=\D(A^*)$, where
\[
  \D(A^*) = \{ u \in H \colon v \mapsto \langle u,Av \rangle_{L^2}
  \text{ is bounded}\} \,.
\]
(I.e., $v \mapsto \langle u,Av \rangle_{L^2}$ can be represented by
$\langle f,v \rangle_{L^2}$ by the Riesz representation theorem.)

\item Show that $A^{-1}$ is a compact operator on $H$.
\end{enumerate}

\item Due to the results in the previous question, there exists a
complete $L^2$-orthonormal sequence $\{w_j\}$ of eigenfuctions of $A$
with corresponding eigenvalues $\lambda_j$, where $0 < \lambda_1 <
\lambda_2 \leq \lambda_3 \leq \dots$ and $w_j \in \D(A)$ for $j \in
\mathbb N$.  

Thus, if
\[
  u = \sum_{j=1}^\infty u_j \, w_j \,,
\]
we can define $A^\alpha$ for any $\alpha \in \R$ by
\[
  A^\alpha u = \sum_{j=1}^\infty \lambda_j^\alpha \, u_j \, w_j \,,
\]
\[
  \langle u,v \rangle_{\D(A^\alpha)}
  = \sum_{j=1}^\infty \lambda^{2\alpha} \, u_j \, v_j 
  \quad \text{when } 
  u = \sum_{j=1}^\infty u_j \, w_j 
  \text{ and }
  v = \sum_{j=1}^\infty u_j \, v_j 
\]
with domain 
\[
  \D(A^\alpha) = \{ u \in H \colon \lVert u \rVert_{\D(A^\alpha)}^2 =
  \langle u, u \rangle_{\D(A^\alpha)} < \infty \} \,.
\]

Show that $D(A^{1/2}) = V$.

\item In the notation above, let $u \in V$.  Prove that, for $n \leq
3$, there exists a constant $c$ such that
\[
  \lVert u \cdot \nabla u \rVert_{V^*}
  \leq c \, \lVert u \rVert_V^{\tfrac{n}2} \, \
            \lVert u \rVert_H^{2-\tfrac{n}2} \,.
\]

\emph{Note:} Due to the result in the previous question, we can
identify $V^*=\D(A^{-1/2})$, so that 
\[
  \lVert v \rVert_{V^*}
  = \lVert A^{-1/2} v \rVert_{L^2}
  \quad \text{and} \quad
  \langle v, w \rangle_{V^*,V} 
  = \langle A^{-1/2} v, A^{1/2} w \rangle_{L^2} \,.
\]
You will also need the Sobolev inequality in the form given as
equation (14) in the proof of Theorem 5.6.1 in Evans (you can also
quote the result from other sources where the exponents are more
explicit). 

\item In the notation above, suppose that
\begin{gather*}
  u_m \rightharpoonup 0 
  \qquad \text{weakly in $L^2((0,T);V)$} \,, \\
  u_m' \rightharpoonup 0 
  \qquad \text{weakly in $L^p((0,T);V^*)$} \,.
\end{gather*}
for some $p>1$.

\begin{enumerate}
\item Prove that 
\[
  u_m \to 0 
  \qquad \text{in $C([0,T];V^*)$} \,.
\]
\item Then conclude that 
\[
  u_m \to 0
  \qquad \text{in $L^2((0,T);H)$} \,.
\]
\end{enumerate}
\emph{Hint:} For part (a), use the fundamental theorem of calculus,
estimate in the $V^*$-norm, and employ an argument similar to the one
given in part (1) of the proof of Evans, Theorem 5.9.4 on p.~287.

You will also need to refer to the compactness of the embedding $V
\subset H$.

\end{enumerate}


\end{document}

