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Section 9.2 Banach-Space Valued Functions

Banach-space valued function become important when we solve partial differential equations (PDEs). As an example let us look at a reaction diffusion equation for a physical or biological quantity \(u(x,t)\text{:}\)
\begin{equation} \frac{\partial u}{\partial t} = D\Delta u + f(u).\label{RDE}\tag{9.2.1} \end{equation}
If \(u(x,t)\) denotes the solution, then for each time \(t\geq 0\) we have a function of space \(x\text{:}\) \(u(t): x\mapsto u(x,t)\text{,}\) hence this map is in some functions space and we write \(u(t)\in X\) for all \(t\geq 0\text{.}\) Typical function spaces in the context of PDEs are given below.

For example \(X=C^0([0,T], L^p(\Omega))\) has the norm
\begin{equation*} \|u\|_X = \sup_{0\leq t\leq T} \|u(t)\|_{L^p(\Omega)}. \end{equation*}

The space \(X=L^p([0,T], L^q(\Omega)) \) has the norm
\begin{equation*} \|u\|_X = \left(\int_0^T \|u(t)\|_{L^q(\Omega)}^p dt\right)^{\frac{1}{p}}. \end{equation*}
The norm in \(L^p(0,T; L^p(\Omega))\) can be written in two ways:
\begin{equation*} \|u\|_X = \left(\int_0^T \|u(t)\|_p^p dt\right)^{\frac{1}{p}} = \left(\int_0^T\int_\Omega |u(x,t)|^p dx dt \right)^{\frac{1}{p}} = \|u\|_{L^p([0,T]\times\Omega)} . \end{equation*}

The space \(X=L^2([0,T], L^2(\Omega))\) is a Hilbert space with inner product
\begin{equation*} \langle u,v\rangle_X = \int_0^T\int_\Omega u(x,t) v(x,t) dx dt. \end{equation*}
There is a Fundamental Theorem of Calculus for Banach-space valued functions.
If \(p=\infty\) then \(u\in W^{1,\tilde p}([0,T],B)\) for some \(\tilde p<\infty\text{.}\) Hence we consider \(p<\infty\) right away. We mollify \(u\) to be equal to zero outside of the interval \([0,T]\) and call the mollification \(u_h\text{.}\) It is easy to check that
\begin{equation*} \frac{d u_h(t)}{dt} = \left(\frac{du(t)}{dt}\right)_h \end{equation*}
and
\begin{equation*} u_h\to u, \qquad\mbox{and}\qquad \frac{du_h}{dt} \to \frac{du}{dt} \qquad \mbox{in }\quad L^p \qquad \mbox{for } h\to 0. \end{equation*}
For \(h>0\) the function \(u_h\) is continuously differentiable and we use the Fundamental Theorem of Calculus:
\begin{equation*} u_h(t) = u_h(s) +\int_s^t \frac{d u_h}{dt}(\tau) d\tau . \end{equation*}
Taking the limit as \(h\to 0\) gives equation (9.2.2).

For the sup-norm estimate we chose \(t=0\) and \(s=t\) and find
\begin{equation*} \|u(0)\|_B \leq \|u(t)\|_B +\int_0^t \|u'(\tau)\|_B d\tau, \end{equation*}
which integrated from \(0\) to \(T\) becomes
\begin{align*} T\|u(0)\|_B \amp\leq\amp \int_0^T \|u(t)\|_B dt + \int_0^T \int_0^t \|u'(\tau)\|_B d\tau dt\\ \amp\leq \amp \|u\|_{L^1([0,T],B)} + T\|u'\|_{L^1([0,T],B)}\\ \amp\leq \amp T^{\frac{1}{q}}\|u\|_{L^p([0,T],B)} + T^{\frac{1+q}{q}} \|u'\|_{L^p([0,T],B)}, \end{align*}
where we used H\"olders inequality and the fact that \(\frac{1+q}{q} = \frac{1}{q} +1 \) in the last step. We use (9.2.2) again and the previous estimates to get
\begin{align*} \|u(t)\|_B \amp\leq \amp \|u(0)\|_B +\int_0^t \|u'(\tau)\|_B d\tau\\ \amp\leq \amp \|u(0)\|_B +T^{\frac{1}{q}}\|u'\|_{L^p([0,T],B)}\\ \amp\leq\amp T^{\frac{1-q}{q}} \|u\|_{L^p([0,T],B)} + 2 T^{\frac{1}{q}} \|u'\|_{L^p([0,T],B)}\\ \amp\leq\amp C \|u \|_{W^{1,p}([0,T],B), } \end{align*}
which proves the last estimate of the Theorem.
If we apply Theorem 9.2.1 to reaction diffusion equations
\begin{equation*} u_t = D\Delta u + f(u) \end{equation*}
we often have that
\begin{equation*} u\in L^2([0,T], H_0^1(\Omega))\quad\mbox{ and } \quad u_t \in L^2([0,T], H^{-1}). \end{equation*}
Since \(H^{1}\subset H^{-1}\) it follows that
\begin{equation*} u\in H^1([0,T], H^{-1}) . \end{equation*}
Then, by the above Theorem 9.2.1 we find
\begin{equation*} u\in C^0([0,T]; H^{-1}). \end{equation*}
In this context, the reaction diffusion equation presents itself as an equality in \(H^{-1}\text{:}\)
\begin{equation*} \underbrace{u_t}_{H^{-1}} = \underbrace{D\Delta \underbrace{u}_{H^1}}_{H^{-1}} + \underbrace{f(u)}_{H^{-1}}. \end{equation*}
Of course, we made no assumptions here on the growth term \(f(u)\text{.}\) But it is clear what the assumptions of \(f\) should be. One possibility is to assume that the map \(f: H^1\to H^{-1}\) is globally Lipschitz continuous.
We cite one more result in this context without giving a proof. The proof can be found in Robinson page 214 ff. \cite{robinson}.
The previous theorem opens the door for numerical approximations of partial differential equations such as the Galerkin Method. For the Galerkin method we consider a finite set of \(n\) orthogonal basis functions, for example sine and cosine functions, and project the differential equation onto the subspace spanned by those basis functions. Since this subspace is finite dimensional, the projection gives us an \(n\)-dimensional ordinary differential equation (ODE). We call the solution \(u_n\text{.}\) We are often able to show that
\begin{equation*} u_n\subset L^2([0,T], H_0^1) \qquad u_{n,t} \subset L^2([0,T], H^{-1}) \end{equation*}
is uniformly bounded in those spaces. Then chosing \(X=H_0^1, H=L^2, Y= H^{-1}\) the previous Theorem provides us with a convergent subsequence
\begin{equation*} u_{n_k} \to u \qquad \mbox{in} \qquad L^2((0,T)\times \Omega) . \end{equation*}
This limit is then a candidate for a solution of the original reaction diffusion equation. This is the essential step of the Galerkin method. However, the set of basis functions need to be chosen carefully. Also, more work is needed to show that the time derivative converges, and that the limit function is continuous in the right spaces. Here we refer to other textbooks such as \cite{robinson}, where the entire proof of the Galerkin method occupies about 6 pages. We develop a simpler case in Exercise 6.4.3.