Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help.
Let's assume we have a function $$ \mathbf{u} (\mathbf{x}; t) = \left( u_{1}, u_{2}, u_{3} \right) \left( x_{1},x_{2},x_{3}; t\right) : \Omega \times [ 0,T] \to \mathbb{R}^{3}\quad \Omega \subset \mathbb{R}^{3}, $$ where
- $\mathbf{u}(\mathbf{x},t)\in L^{\infty}(0,T, \mathbf{H}^{1}(\Omega)) \cap L^{2}(0,T,\mathbf{V}_{\mathbf{g}}(\Omega)) $, and
- $\mathbf{V}_{\mathbf{g}}(\Omega) = \left\lbrace \mathbf{u} \in \mathbf{H}^{2} (\Omega) : \mathbf{u}|_{\partial \Omega} = \mathbf{g}, \ \nabla\cdot \mathbf{u} = 0 \ \mathrm{ in} \ \Omega \right\rbrace$.
Is the following expression of the norm correct? \begin{align*} \Vert \mathbf{u} \Vert_{L^{2} \left(0,T, \mathbf{H}^{2}(\Omega) \right)} &= \left( \int_{0}^{T} \Vert \mathbf{u} \Vert^{2}_{L^{2}(\Omega)} + \Vert \mathbf{u}^{\prime} \Vert^{2}_{L^{2}(\Omega)} + \Vert \mathbf{u}^{\prime \prime} \Vert^{2}_{L^{2}(\Omega)} d t \right)^{1/2} , \\ \Vert \mathbf{u} \Vert^{2}_{L^{2}(\Omega)} &= \left\langle \mathbf{u},\mathbf{u} \right\rangle_{L^{2}(\Omega)} \\ &= \left\langle \left( u_{1}, u_{2}, u_{3} \right),\left( u_{1}, u_{2}, u_{3} \right) \right\rangle_{L^{2}(\Omega)} \\ &= \iiint_{\Omega} \sum_{i = 1}^{3} \vert u_{i} \vert^{2} d \Omega \\ &= \Vert u_{1} \Vert^{2}_{L^{2}(\Omega)} + \Vert u_{2} \Vert^{2}_{L^{2}(\Omega)} + \Vert u_{3} \Vert^{2}_{L^{2}(\Omega)} , \\ \Vert \mathbf{u}^{\prime} \Vert^{2}_{L^{2}(\Omega)} &= \left\langle \mathbf{u}^{\prime},\mathbf{u}^{\prime} \right\rangle_{L^{2}(\Omega)} \\ &= \left\langle \left( u^{\prime}_{1}, u^{\prime}_{2}, u^{\prime}_{3} \right),\left( u^{\prime}_{1}, u^{\prime}_{2}, u^{\prime}_{3} \right) \right\rangle_{L^{2}(\Omega)} \\ &= \iiint_{\Omega} \left| u^{\prime}_{1} \right|^{2} + \left| u^{\prime}_{2} \right|^{2} + \left| u^{\prime}_{3} \right|^{2} d \Omega \\ &= \iiint_{\Omega} \left| u^{\prime}_{1} \right|^{2} d \Omega + \iiint_{\Omega} \left| u^{\prime}_{2} \right|^{2} d \Omega + \iiint_{\Omega} \left| u^{\prime}_{3} \right|^{2} d \Omega \\ &= \left\langle u^{\prime}_{1} , u^{\prime}_{1} \right\rangle + \left\langle u^{\prime}_{2} , u^{\prime}_{2} \right\rangle + \left\langle u^{\prime}_{3} , u^{\prime}_{3} \right\rangle \\ &= \left\langle \left( \frac{\partial u_{1}}{\partial x_{1}}, \frac{\partial u_{1}}{\partial x_{2}}, \frac{\partial u_{1}}{\partial x_{3}} \right), \left( \frac{\partial u_{1}}{\partial x_{1}}, \frac{\partial u_{1}}{\partial x_{2}}, \frac{\partial u_{1}}{\partial x_{3}} \right) \right\rangle + \left\langle \left( \frac{\partial u_{2}}{\partial x_{1}}, \frac{\partial u_{2}}{\partial x_{2}}, \frac{\partial u_{2}}{\partial x_{3}} \right), \left( \frac{\partial u_{2}}{\partial x_{1}}, \frac{\partial u_{2}}{\partial x_{2}}, \frac{\partial u_{2}}{\partial x_{3}} \right) \right\rangle \\ & \quad + \left\langle \left( \frac{\partial u_{3}}{\partial x_{1}}, \frac{\partial u_{3}}{\partial x_{2}}, \frac{\partial u_{3}}{\partial x_{3}} \right), \left( \frac{\partial u_{3}}{\partial x_{1}}, \frac{\partial u_{3}}{\partial x_{2}}, \frac{\partial u_{3}}{\partial x_{3}} \right) \right\rangle \\ &= \Vert u^{\prime}_{1} \Vert^{2}_{L^{2}(\Omega)} + \Vert u^{\prime}_{2} \Vert_{L^{2}(\Omega)} + \Vert u^{\prime}_{3} \Vert^{2}_{L^{2}(\Omega)} \\ &= \iiint_{\Omega} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \left| \frac{\partial u_{i}}{\partial x_{j}} \right|^{2} d \Omega, \end{align*} \begin{align*} \Vert \mathbf{u}^{\prime \prime} \Vert^{2}_{L^{2}(\Omega)} &= \left\langle\mathbf{u}^{\prime \prime},\mathbf{u}^{\prime \prime} \right\rangle_{L^{2}(\Omega)} \\ &= \left\langle \left( u^{\prime \prime}_{1}, u^{\prime \prime}_{2}, u^{\prime \prime}_{3} \right),\left( u^{\prime \prime}_{1}, u^{\prime \prime}_{2}, u^{\prime \prime}_{3} \right) \right\rangle_{L^{2}(\Omega)} \\ &= \iiint_{\Omega} \left| u^{\prime \prime}_{1} \right|^{2} + \left| u^{\prime \prime}_{2} \right|^{2} + \left| u^{\prime \prime}_{3} \right|^{2} d \Omega \\ &= \iiint_{\Omega} \left| u^{\prime \prime}_{1} \right|^{2} d \Omega + \iiint_{\Omega} \left| u^{\prime \prime}_{2} \right|^{2} d \Omega + \iiint_{\Omega} \left| u^{\prime \prime}_{3} \right|^{2} d \Omega \\ &= \left\langle u^{\prime \prime}_{1} , u^{\prime \prime}_{1} \right\rangle + \left\langle u^{\prime \prime}_{2} , u^{\prime \prime}_{2} \right\rangle + \left\langle u^{\prime \prime}_{3} , u^{\prime \prime}_{3} \right\rangle \\ &= \Bigg< \left( \frac{\partial^{2} u_{1}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{1}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{1}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{1}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{1}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{1}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{3} \partial x_{1}} \right), \\ & \quad \left( \frac{\partial^{2} u_{1}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{1}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{1}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{1}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{1}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{1}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{1}}{\partial x_{3} \partial x_{1}} \right) \Bigg> \\ & \quad + \Bigg< \left( \frac{\partial^{2} u_{2}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{2}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{2}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{2}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{2}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{2}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{3} \partial x_{1}} \right), \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \frac{\partial^{2} u_{2}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{2}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{2}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{2}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{2}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{2}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{2}}{\partial x_{3} \partial x_{1}} \right) \Bigg> \\ & \ \ \ \ + \Bigg< \left( \frac{\partial^{2} u_{3}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{3}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{3}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{3}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{3}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{3}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{3} \partial x_{1}} \right), \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \frac{\partial^{2} u_{3}}{\partial x^{2}_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{1} \partial x_{2}}, \frac{\partial u^{2}_{3}}{\partial x_{1} \partial x_{3}}, \frac{\partial^{2} u_{3}}{\partial x^{2}_{2}}, \frac{\partial^{2} u_{3}}{\partial x_{2} \partial x_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{2} \partial x_{3}}, \frac{\partial^{2} u_{3}}{\partial x^{2}_{3}}, \frac{\partial^{2} u_{3}}{\partial x_{3} \partial x_{1}}, \frac{\partial^{2} u_{3}}{\partial x_{3} \partial x_{1}} \right) \Bigg> \\ &= \Vert u^{\prime \prime}_{1} \Vert^{2}_{L^{2}(\Omega)} + \Vert u^{\prime \prime}_{2} \Vert_{L^{2}(\Omega)} + \Vert u^{\prime \prime}_{3} \Vert^{2}_{L^{2}(\Omega)} \\ &= \iiint_{\Omega} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \left| \frac{\partial^{2} u_{i}}{\partial x^{2}_{j}} \right|^{2} + 2 \sum_{i = 1}^{3} \left| \frac{\partial^{2} u_{i}}{\partial x_{1} \partial x_{2}} \right|^{2} + 2 \sum_{i = 1}^{3} \left| \frac{\partial^{2} u_{i}}{\partial x_{2} \partial x_{3}} \right|^{2} d \Omega. \end{align*}
