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\def\R{\mathbb{R}}
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\begin{document}

\title{Partial Differential Equations}
\author{Homework 6}
\date{due May 15, 2014}
\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 $W_t = \phi_t(W)$.

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 \emph{Setting:} 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 $H^1$.  As is shown in Constantin \& Foias, for
example, 
\[
  H = \{ u \in L^2(U,\R^n) \colon \nabla \cdot u = 0 \text{ and }
  \gamma(u)=0 \} 
\]
where the divergence understood in the sense of weak derivatives and
where $\gamma$ is the normal trace operator satisfying $\gamma(u)=\nu
\cdot u$ if $u \in C^\infty(\overline U)$ which is a bounded operator
from
\[
  E(U) = \{ u \in L^2(U,\R^n) \colon \nabla \cdot u \in L^2(U,\R) \} 
\]
to $H^{-1/2}(U)$.  It is further known that
\[
  V = \{ u \in H_0^1(U,\R^n) \colon \nabla \cdot u = 0 \} \,.
\]

Recall further that the \emph{Stokes operator} $A \colon \D(A) \to H$
is defined by $A = - P\Delta$ where $P$ is the $L^2$-orthogonal
projector onto $H$ and $\D(A) = H^2(U,\R^n) \cap V$.  It is a positive
self-adjoint operator with compact inverse.  Hence, 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 
\[
  \D(A^\alpha) = \{ u \in H \colon \lVert u \rVert_{\D(A^\alpha)}^2 =
  \langle u, u \rangle_{\D(A^\alpha)} < \infty \} \,.
\]
Then, in particular, $V = D(A^{1/2})$ and, writing $V^*$ for the
$L^2$-dual of $V$, $V^*=\D(A^{-1/2})$ so that
\[
  \lVert v \rVert_{V^*}
  = \lVert A^{-1/2} v \rVert_{L^2}
  \quad \text{and} \quad
  \langle v, w \rangle_{V^*,V} 
  = \langle A^{-1/2} v, A^{1/2} w \rangle_{L^2} \,.
\]

\emph{Problem:} In the setting 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{Hint:} You will 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 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}


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