Relation between distributional derivatives
Relation between distributional derivatives in \(L^2(0,T;L^2(\Omega))\) and \(L^2(Q)\)
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 \(\|u_{x_i}(t)\|_{H^{-1}}\in L^2(0,T)\).
Indeed, given \(v\in H_0^1(\Omega)\),\[ \begin{equation} \left|\langle u_{x_i}(t),v\rangle\right|\leq |(u(t),v_x)|\leq \|u(t)\|_{L^2(\Omega)} \|v\|_{H_0^1(\Omega)}, \end{equation} \]
whence,\[ \|u_{x_i}(t)\|_{H^{-1}(\Omega)}\leq \|u(t)\|_{L^2(\Omega)}\in L^2(0,T). \]
\(\square\)
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.
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, e 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, \[z(t,\cdot)\varphi(t) \in H_0^1(\Omega),\ \forall \varphi\in \mathcal{D}(0,T) \text{ and a.e in } (0,T).\]