To analyse fluid motions, we seek expressions for the commutator of the Lagrange-Laplace transform and various spatial differential operators.
For a function $f(\mathbf{r},t)$ of space and time, we define two distinct Laplace transforms. The Euler-Laplace transform (ELT) is evaluated at a fixed point in space \begin{equation} \widehat{f}(\mathbf{r}_0,s) \equiv {\cal L}[f](\mathbf{r}_0,s) = \int_{0\ {\cal C}_0}^\infty e^{-st} f(\mathbf{r},t)\,\mathrm{d} t \,. \label{eq:LT} \end{equation} Here, ${\cal C}_0$ is a line in the $\mathbf{r}$-$t$ space parallel to the time axis and passing through the point $(\mathbf{r}_0,0)$.
The Lagrange-Laplace transform (LLT) is evaluated along a trajectory of the motion: \begin{equation} \widetilde{f}(\mathbf{r}_0,s) \equiv \mathfrak{L}[f](\mathbf{r}_0,s) = \int_{0\ {\cal C}_1}^\infty e^{-st} f(\mathbf{r}(t),t)\,\mathrm{d} t \,. \label{eq:LL} \end{equation} Here, ${\cal C}_1$ is the trajectory of the motion starting at the point $(\mathbf{r}_0,0)$.
Non-commutativity of the LLT
We aim to solve various approximate forms of the Navier-Stokes equations governing fluid flow. We consider the momentum equation in vector form, $$ \frac{\mathrm{d}\mathbf{V}}{\mathrm{d} t} + \mathbf{\mathbf{\nabla}}\Phi = \mathbf{0} \,. $$ We define the Lagrange-Laplace Transform $\mathfrak{L}$ along the trajectory $\mathcal{C}$ that passes through the initial point $\mathbf{r}_0$: $$ \mathfrak{L} \left\lbrace \mathbf{V}\right\rbrace = \int_{0\, \mathcal{C}}^{\infty} \mathbf{V}\, \mathrm{e}^{-st}\mathrm{d} t \,. $$
Applying $\mathfrak{L}$ to the momentum equation, we get $$ \mathfrak{L}\left\lbrace \frac{\mathrm{d}\mathbf{V}}{\mathrm{d} t}\right\rbrace + \mathfrak{L}\left\lbrace \mathbf{\mathbf{\nabla}}\Phi \right\rbrace = \mathbf{0} \,. $$ We require an equation connecting the transformed variables. The first term is $\mathfrak{L}\{\mathrm{d}\mathbf{V}/ \mathrm{d} t \} = s\widehat{\mathbf{V}}-\mathbf{V}(0)$. In the second, we need to replace $\mathfrak{L}\left\lbrace \mathbf{\mathbf{\nabla}}\Phi \right\rbrace$ by $\mathbf{\mathbf{\nabla}}\mathfrak{L}\left\lbrace \Phi \right\rbrace$.
The Euler-Laplace transform $\cal L$ commutes with spatial operators like $\partial_x$ and $\nabla^2$. This is not the case for the Lagrange-Laplace transform $\mathfrak{L}$. We write $$ [ \mathfrak{L}, \mathbf{\nabla} ] \equiv \mathfrak{L}\mathbf{\nabla} - \mathbf{\nabla}\mathfrak{L} \,. $$ and, to replace $\mathfrak{L}\mathbf{\nabla}$ by $\mathbf{\nabla}\mathfrak{L}$, we require an expression for the commutator. Can quantities like $[ \mathfrak{L}, \mathbf{\nabla} ]$ and $[ \mathfrak{L}, \nabla^2 ]$ be expressed in analytical form?