1. Distributional Derivatives

Given \(T>0\) and \(\Omega\subset \mathbb{R}^n\) an open set let us denote \(Q=(0,T)\times \Omega\). A function \(u\in L^2(Q)\) can also be seen as a vector valued mapping in \(L^2(0,T;L^2(\Omega))\). This give us two notion of distributional derivatives. Our aim is prove they coincide. In order to do that, we need to introduce a fundamental density result given in (Friedlander et al. 1998).

Given \(\Omega_x\subset \mathbb{R}^n\), \(\Omega_y\in \mathbb{R}^m\) open sets, let \(\mathcal{D}(\Omega_x)\), \(\mathcal{D}(\Omega_y)\) and \(\mathcal{D}(\Omega_x\times \Omega_y)\) be the test space of infinitely differentiable functions with compact support.

Definition 1.1: Let \(f:\Omega_x\longrightarrow \mathbb{R}\) and \(g:\Omega_y\longrightarrow \mathbb{R}\). The Tensor product of \(f\) and \(g\) is a function on \(\Omega_x\times \Omega_y\) defined by \[f\otimes g(x,y)=f(x)g(y),\ \forall (x,y)\in \Omega_x\times\Omega_y.\]

Let us denote by \(\mathcal{D}(\Omega_x)\otimes \mathcal{D}(\Omega_y)\) the subspace of \(C_c^\infty(\Omega_x\times\Omega_y)\) generated by function of the form \(\phi\otimes \psi\), where \(\phi \in \mathcal{D}(\Omega_x)\) and \(\psi \in \mathcal{D}(\Omega_y)\).

Theorem 1.2: The space \(\mathcal{D}(\Omega_x)\otimes \mathcal{D}(\Omega_y)\) is dense in \(\mathcal{D}(\Omega_x\times \Omega_y)\).

proof: See Theorem 4.3.1 (Friedlander et al. 1998).

Now, let us define the two notions of distributional derivatives. Let us denote by \(((\cdot,\cdot))\) inner product in \(L^2(Q)\) and \((\cdot,\cdot)\) the inner product in \(L^2(\Omega)\) or \(L^2(0,T)\).

Given \(u\in L^2(Q)\) we have distributional derivatives \(D_tu,D_xu\in \mathcal{D}'(Q)\) given by

\[\begin{split} &\langle\!\langle D_tu, \phi \rangle\!\rangle =(\!(u,\phi_t)\!)=-\iint\limits_Q u(t,x)\phi_t(t,x)\,dx dt,\ \forall \phi \in \mathcal{D}(Q), \\ & \langle\!\langle D_{x_i}u, \phi\rangle\!\rangle= (\!(u,\phi_{x_i})\!)=-\iint\limits_Q u(t,x)\phi_{x_i}(t,x)\,dxdt\ \forall \phi \in \mathcal{D}(Q),\ \forall i=1,\ldots,n. \end{split}\]

We also can see \(u\) as a vector-valued mapping in \(L^2(0,T;L^2(\Omega))\), that is, \(u: t\in (0,T)\longmapsto u(t,\cdot) \in L^2(\Omega)\). In this space, we have the following notion of distributional derivative: \(u':\mathcal{D}(0,T)\longrightarrow L^2(\Omega)\) is defined by

\[\langle u',\varphi \rangle=-(u,\varphi')=-\int_0^T u(t,x)\varphi(t)d t, \ \forall \varphi \in \mathcal{D}(0,T).\]

Similarly, we can define \(u_{x_i}:(0,T)\longrightarrow \mathcal{D}'(\Omega)\) defined by

\[\langle u_{x_i}(t),\psi\rangle=-(u(t),\psi_{x_i})=-\int\limits_\Omega u(t,x)\psi_{x_i}(x)\,dx, \forall \psi \in \mathcal{D}(\Omega).\]

Proposition 1.3: If \(u\in L^2(0,T;L^2(\Omega))\), then \(u_{x_i}\in L^2(0,T;H^{-1}(\Omega))\).

Proof: Since \(\mathcal{D}(\Omega)\) is dense in \(H_0^{1}(\Omega)\), we can extend \(u_{x_i}(t)\) to \(H^{-1}(\Omega)\). We just need to prove that \(\|u_{x_i}(t)\|_{H^{-1}}\in L^2(0,T)\).

\[ \begin{equation} \left|\langle u_{x_i}(t),v\rangle\right|\leq |(u(t),v_x)|\leq \|u(t)\|_{L^2(\Omega)} \|v\|_{H_0^1(\Omega)}, \end{equation} \]

\[ \|u_{x_i}(t)\|_{H^{-1}(\Omega)}\leq \|u(t)\|_{L^2(\Omega)}\in L^2(0,T). \]

\(\square\)

Our aim is to prove that the two notions of distributional derivative coincide. To do that, let us see define \(u'\) and \(u_{x_i}\) as distributions in \(\mathcal{D}(Q)\).

Given \(\varphi\in \mathcal{D}(0,T)\) and \(\psi\in \mathcal{D}(\Omega)\), we define

\[\begin{equation} \langle\!\langle u',\varphi\otimes\psi\rangle\!\rangle :=(\langle u',\varphi\rangle,\psi)=-\int\limits_\Omega (u,\varphi')\psi\, dx=-\iint\limits_Q u(t,x)\varphi'(t)\psi(x)\,dxdt=\langle\!\langle D_tu,\varphi\otimes \psi\rangle\!\rangle. \end{equation}\]

Since \(\mathcal{D}(0,T)\otimes \mathcal{D}(\Omega)\) is dense in \(\mathcal{D}(Q)\) (Theorem 1.2) we can define \(u'\) as a distribution in \(\mathcal{D}(Q)\) and we also have that \(u'=D_tu\in \mathcal{D}'(Q)\).

Similarly, we define

\[\begin{equation*} \langle\!\langle u_{x_i},\varphi\otimes\psi\rangle\!\rangle:=(\langle u_{x_i}(t),\psi\rangle,\varphi)=-\int_0^T (u,\psi_{x_i})\varphi\, dx=-\iint\limits_Q u(t,x)\varphi(t)\psi_{x_i}(x)\, dxdt=\langle\!\langle D_xu,\varphi\otimes \psi\rangle\!\rangle. \end{equation*}\]

And, we have that \(u_{x_i}=D_{x_i}u\) in \(\mathcal{D}'(Q)\).

Therefore, we also can conclude that \(D_tu=u'\in D'(0,T;L^2(\Omega))\) and \(D_{x_i}u=u_{x_i}\in L^2(0,T;H^{-1}(\Omega))\).

Furthermore, we can see that \(u_{x_ix_j}\in L^2(0,T;H^{-2}(\Omega))\). Following the same reasoning, we can define \(u_{x_ix_j}\) on \(\mathcal{D}(0,T)\otimes \mathcal{D}(\Omega)\) and, by a density argument, extend it to \(\mathcal{D}(Q)\) with \(D_{x_ix_j}u=u_{x_ix_j}\) in \(\mathcal{D}'(Q)\). Similarly, \(u_{tt}\in \mathcal{D}'(0,T;L^2(\Omega))\), then we can define \(u''\) as a distribution \(\mathcal{D}'(Q)\) with \(D_{tt}u=u''\).

Therefore, from now on, we will stop using the \(D\)-notation for distributional derivative in \(Q\) and use the same notation of vector-distribution.

2. Application to ultra-weak solution of wave equation

Let us consider the problem

\[ \begin{equation}\tag{1}\label{pb1} \begin{cases} z''-\Delta z=0 \text{ in } Q,\\ z=0 \text{ on } (0,T)\times \partial \Omega,\\ z(0)=z^0,\ z'(0)=z^1 \text{ in } \Omega, \end{cases} \end{equation} \]

where \(z^0\in L^2(\Omega)\) and \(z^1\in H^{-1}(\Omega)\). As the initial values are not regular, we need a different definition of solution, the so called solution by transposition or ultra weak solution.

Definition 2.1: Given \((z^0,z^1)\in L^2(\Omega)\times H^{-1}(\Omega)\), we say \(z\in L^2(Q)\) is a ultra weak solution or a solution by transposition of \(\eqref{pb1}\) if, for each \(f\in \mathcal{D}(Q)\) given, we have \[ \iint\limits_Q zf d xd t=-(z^0,\theta'(0))+\langle z^1,\theta(0)\rangle, \]

where \(\theta\) is the classical solution of

\[ \begin{equation}\label{pb2}\tag{2} \begin{cases} \theta''-\Delta \theta=f \text{ in } Q,\\ \theta=0 \text{ on } (0,T)\times \partial \Omega,\\ \theta(T)=\theta'(T)=0\text{ in } \Omega. \end{cases} \end{equation} \]

Let us estate an existence result which can be seen in Theorem 4.2, pag. 46 of (Lions 1988) or Theorem 4.1, pag. 45 of (Medeiros, Miranda, and Lourêdo 2013)

Theorem 2.2: Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with \(\partial \Omega\) of class \(C^2\). For all \((z^0,z^1)\in L^2(\Omega)\times H^{-1}(\Omega)\), there exist a unique ultra weak solution \(z\) of \(\eqref{pb1}\). Moreover, \(z\in C^0([0,T];L^2(\Omega))\cap C^1([0,T];H^{-1}(\Omega))\) and there exists \(C>0\) such that \[ \|z\|_{L^\infty(0,T;L^2(\Omega))}+\|z'\|_{L^\infty(0,T;L^2(\Omega))} \leq C\left(\|z^0\|_{L^2(\Omega)}+\|z^1\|_{H^{-1}(\Omega)}\right). \]

For the sake of simplicity, from now on, we will consider \(\Omega=(0,1)\).

Given \(z^1\in H^{-1}(\Omega)\), take \(\psi\in H_0^1(\Omega)\) the weak solution to the elliptic problem \[\begin{equation*} \begin{cases} \psi_{xx} = z^1 \text{ in } \Omega,\\ \psi=0 \text{ in } \partial \Omega. \end{cases} \end{equation*}\]

Given \(z\) a ultra weak solution of \(\eqref{pb1}\), define \(w(t,x)=\int_0^t z(s,x)d s+\psi(x)\). Let us prove that \(w\) is a weak solution of

\[\begin{equation}\label{pb3}\tag{3} \begin{cases} w''-w_{xx}=0 \text{ in } Q,\\ w=0 \text{ on } (0,T)\times \partial \Omega,\\ w(0)=\psi,\ w'(0)=z^0 \text{ in } \Omega. \end{cases} \end{equation}\]

It is easy to see that \(w(0)=\psi\in H_0^1(\Omega)\), \(w'(0)=z^0\in L^2(\Omega)\) and \(w=0\) on \((0,T)\times \Omega\). We just need to prove that \(w\) satisfies the equation in the weak sense.

First, let us prove that \[z''-z_{xx}=0 \text{ in } \mathcal{D}'(Q).\] Indeed, given \(\phi\in \mathcal{D}(Q)\), we have that \(\phi\) is the solution to \(\eqref{pb2}\) with \(f=\phi''-\phi_{xx}\). From definition of ultra weak solution we have that \[\iint\limits_Q z(\phi''-\phi_{xx})d xd t=-(z^0,\phi'(0))+\langle z^1,\phi(0)\rangle =0.\] Hence, \[\langle\!\langle z''-z_{xx},\phi\rangle\!\rangle=0,\] that is \[z''-z_{xx} \text{ in } \mathcal{D}'(Q).\] Consequently, \[z''-z_{xx}=0 \text{ a.e in } Q.\] Since \(z_{xx}\in C^0([0,T];H^{-2}(\Omega))\), then \(z''\in C^0([0,T];H^{-2}(\Omega))\). Also, note that \(w''=z'\in C^0([0,T];H^{-1}(\Omega))\). Hence, we can see \[w_{xx}(t)=\int_0^tz_{xx}(s)d s+\psi_{xx}=\int_0^t z''(s)d s+z^1=z'(t)-z'(0)+z^1=w''(t).\] Then \(w\) is a weak solution to \(\eqref{pb3}\) and \(w_{xx}\in C^0([0,T];H^{-1}(\Omega))\). Hence, from regularity results, we know that \(w\in L^\infty(0,T;H_0^1(\Omega))\), which implies \(z=w'\in D'(0,T;H_0^1(\Omega))\), that is, \[z(t,\cdot)\varphi(t) \in H_0^1(\Omega),\ \forall \varphi\in \mathcal{D}(0,T) \text{ and a.e in } (0,T).\]