Exercises 6.4 Exercises
1. Conserved integral (level 2).
Let \(V\) be the subspace of \(H^1(\Omega)\text{,}\) consisting of functions with zero integral
\begin{equation*}
V:=\left\{u\in H^1(\Omega):\int_\Omega u(x) dx =0\right\}.
\end{equation*}
Arguing by contradiction, use Rellich-Kondrachov compactness to show that there exists a constant \(C>0\) such that we have a Poincar\'e inequality
\begin{equation*}
\|u\|_2\leq C \|\nabla u\|_2.
\end{equation*}
2. Distributions (level 1).
For \(\psi\in C^\infty_c(\Omega)\) and \(u\in D'(\Omega)\) we define a distribution \(u\psi\) by
\begin{equation*}
\langle \psi u, \phi\rangle = \langle u,\psi\phi\rangle, \qquad \mbox{ for all } \qquad \phi\in C^\infty_c(\Omega).
\end{equation*}
Show that indeed \(\psi u\in D'(\Omega)\) and that
\begin{equation*}
D(\psi u) = u D\psi + \psi Du .
\end{equation*}
3. Galerkin method (level 3).
Consider the reaction-diffusion equation on \(\Omega\text{:}\)
\begin{equation}
u_t = \Delta u + \alpha u, u|_{\partial \Omega} = 0,
u(x,0) = u_0(x), \label{sob-eins}\tag{6.4.1}
\end{equation}
with \(u_0\in L^2(\Omega)\text{.}\) Let \(\{\psi_i\}_{i\geq 0}\) denote an ONB of \(L^2(\Omega)\) of eigenfunctions of \(\Delta\) with eigenvalues \(\lambda_i\text{.}\) To construct a solution for (6.4.1), we apply projections \(P_n\) to (6.4.1) and pass to the limit for large \(n\text{:}\) - Apply the projections \(P_n\) to (6.4.1) and argue that the resulting system has a unique solution \(u_n\) for each \(n\geq 1\text{.}\)
- Derive the estimate\begin{equation} \frac{d}{dt}\frac{\|u_n\|_2^2}{2} +\|\nabla u_n\|_2^2 \leq \alpha \|u_n\|_2^2. \label{zwei}\tag{6.4.2} \end{equation}
- Use Gronwall's Lemma to show that for each \(T>0\) \(u_n\) is uniformly bounded in \(L^\infty([0,T], L^2(\Omega))\text{.}\)
- Integrate (6.4.2) from \(0\) to \(T\) and show that \(u_n\) is uniformly bounded in the space \(L^2([0,T], H^1(\Omega))\text{.}\)
- Show that \({u_n}\) has a convergent subsequence with limit \(u\text{.}\) Identify the type of convergence and the function space of the limit function \(u\text{.}\)
- Explain in which sense \(P_{n_j} u_{n_j}\) converges to \(u\text{?}\)
- Explain in which sense \(\Delta u_{n_j}\) converges to \(\Delta u\text{?}\)
- Use (6.4.1) and explain in which sense \(\frac{\partial u_{n_j}}{\partial t} \) converges to \(\frac{\partial u}{\partial t} \text{?}\)
- Conclude that the limit \(u\) is indeed a weak solution of (6.4.1).