Laplace and Fourier Transforms to derive an Unsteady Fundamental Solution

Question

I am considering the equations for unsteady Stokes flow

$$ \left\{ \begin{split} \nabla \cdot u &= 0,\\ \\ \rho \dfrac{\partial u}{\partial t} &= -\nabla p + \mu \nabla^2 u.\\ \end{split} \right. $$

In a paper (which I can provide if necessary) it is stated that one can use Fourier and Laplace transforms on these equations to obtain the unsteady fundamental solution.

I have not used a lot of transform theory before so was wondering how this would work, does one separate the variables out to have an ODE in the time variable which is solved with the Laplace transform and the Fourier transform is kept for the spatial part?

Edit: Obviously one does not transform these equations directly to get the fundamental solution, but these equations with the appropriate singular point forcing.

Answer 1

More a long comment that a full answer.
In some sense your guess is correct, since this is exactly how these transforms are used when they are applied jointly to a PDE: in the first explicit reference I recall, Richard Briggs explicitly says that in his monograph he will "perform Laplace transformation with respect to time and a Fourier transformation with respect to the spatial coordinate ..." ([1], §2,2 p. 12).
However, the reason for performing Laplace transform respect to the time variable for systems of PDEs is subtler respect to the need to solve an (at most second order, in most cases) ordinary differential equation. Indeed, as shown in this Q&A (already cited in comments), if you have a single PDE you can simply apply to it the Fourier transform respect to the spatial variable and solve the resulting ODE by any elementary methods. Now let's consider what happens if you have the following system of first order respect to time PDEs: $$ \partial_t\mathbf u= \mathbf A(\partial_\mathbf{x})\mathbf{u}\label{ex}\tag{Ex.} $$ where

• $\mathbf{u}=(u_1,\ldots,u_n)$, $n\ge 2$ is the unknown $n$-dimensional vector,
• $\mathbf{x}=(x_1,\ldots,x_m)$, $m\ge 1$ is the $m$-dimensional spatial variable $\implies\partial_\mathbf{x}=(\partial_{x_1},\ldots,\partial_{x_m})$ is the $m$-dimensional vector of the partial derivatives respect to the components of $\mathbf{x}$,
• $\mathbf A(\partial_\mathbf{x})$ is a $n\times n$ matrix partial differential operator whose entries are polynomial in the variables $\partial_\mathbf{x}$ with complex coefficients.

If you apply to \eqref{ex} the Fourier transform respect to $\mathbf{x}$ i.e. $\mathscr{F}_{\bf{x}\mapsto\boldsymbol{\xi}}$ you get the following system of ODEs $$ \frac{\mathrm{d}\mathbf u}{\mathrm{d}t}= \mathbf A(2\pi i\boldsymbol{\xi})\mathbf{u}\label{e}\tag{Ex.} $$ which is easily (almost from the theoretic point of view) solvable by calculating its fundamental matrix $$ e^{t\mathbf{A}(2\pi i\boldsymbol{\xi})}\label{fs}\tag{FS} $$ by putting $$ \mathbf{u}=e^{t\mathbf{A}(2\pi i\boldsymbol{\xi})}\mathbf{u}_0 $$ where $\mathbf{u}_0$ is the initial condition for the system \eqref{ex}. However, as Peter Henrici notes is his monumental work [2] §12.5, p. 537 example 7, calculating \eqref{fs} is not an easy task and also can hidden the structure of the solution respect to the spatial variables. Therefore, when dealing with systems of PDEs it is strongly advisable too algebrize completely the problem, i.e. to transform the problem at and in the linear algebra problem of solving possibly determined homogeneous or not, system of linear equations.

In our case, the 3D Stokes system, assuming the notation of [3], pp. 898-899, and putting $\mathbf u=(u,v,w)$, $\mathbf x=(x,y,z)$, $\boldsymbol\xi=(\xi_1, \xi_2,\xi_3)$, we have $$ \left\{ \begin{split} 0 &= \frac{\partial{u}}{\partial x} + \frac{\partial{v}}{\partial y} + \frac{\partial{w}}{\partial z}\\ \rho \dfrac{\partial u}{\partial t} &= -\frac{\partial{p}}{\partial x} + \mu \nabla^2 u + \alpha_1\delta(\mathbf x)\delta(t)\\ \rho \dfrac{\partial v}{\partial t} &= -\frac{\partial{p}}{\partial y} + \mu \nabla^2 v + \alpha_2\delta(\mathbf x)\delta(t)\\ \rho \dfrac{\partial w}{\partial t} &= -\frac{\partial{p}}{\partial z} + \mu \nabla^2 w + \alpha_3\delta(\mathbf x)\delta(t)\\ \end{split} \right.\label{st}\tag{ST} $$ Obviously, since we are dealing with fundamental solutions, we should work in the framework of generalized function, for example distributions: thus we assume that $p, u,v,w$ belong to the space of Schwartz distributions $\mathscr{S}^\prime(\Bbb R^3\times\Bbb R)$, in order to be able to do Fourier analysis.

Applying Laplace transformation $\mathscr{L}_{t\mapsto s}$ we first get $$ \left\{ \begin{split} 0 &= \frac{\partial\hat{u}}{\partial x} + \frac{\partial\hat{v}}{\partial y} + \frac{\partial\hat{w}}{\partial z}\\ \rho s \hat{u} &= -\frac{\partial\hat{p}}{\partial x} + \mu \nabla^2 \hat{u} + \alpha_1\delta(\mathbf x)\\ \rho s \hat{v} &= -\frac{\partial\hat{p}}{\partial y} + \mu \nabla^2 \hat{v} + \alpha_2\delta(\mathbf x)\\ \rho s \hat{w} &= -\frac{\partial\hat{p}}{\partial z} + \mu \nabla^2 \hat{w} + \alpha_3\delta(\mathbf x)\\ \end{split} \right., $$ and then applying the Fourier transform respect to the $\bf{x}$ variable $\mathscr{F}_{\bf{x}\mapsto\boldsymbol{\xi}}$ $$ \left\{ \begin{split} 0 &= \xi_1\hat{u} + \xi_2\hat{v} + \xi_3\hat{w}\\ \rho s \hat{u} &= -2\pi i\xi_1\hat{p} - 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2 \hat{u} + \alpha_1\\ \rho s \hat{v} &= -2\pi i\xi_2\hat{p} - 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2 \hat{v} + \alpha_2\\ \rho s \hat{w} &= -2\pi i\xi_3\hat{p} - 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2 \hat{w} + \alpha_3\\ \end{split} \right.. $$ (by abuse of notation but for simplicity reasons, we do not change the symbols $ \hat{p}, \hat{u}, \hat{v}, \hat{w}$) and thus we finally get $$ \begin{pmatrix} \xi_1 & \xi_2 & \xi_3 & 0 \\ (\rho s + 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2) & 0 & 0 & 2\pi i\xi_1 \\ 0 & (\rho s + 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2) & 0 & 2\pi i\xi_2 \\ 0 & 0 & (\rho s + 4 \mu \pi^2 \Vert\boldsymbol\xi\Vert^2) & 2\pi i\xi_3 \\ \end{pmatrix} \begin{pmatrix} \hat{u}\\ \hat{v}\\ \hat{w}\\ \hat{p} \end{pmatrix}= \begin{pmatrix} 0\\ \alpha_1\\ \alpha_2\\ \alpha_3\\ \end{pmatrix} $$ Now we have a fully algebraic, nonhomogeneous determined linear system which is solvable by elementary means. The solution vector we obtain is the Laplace transform respect to time and the Fourier transform respect to the spatial variable of the fundamental solution of the Stokes system \eqref{st}: and in order to reconstruct the fundamental solution, we shall simply component-wise inverse transform the found algebraic expressions with the aid of tables which, even if it is not the easiest task around, it is nevertheless less daunting that calculating first \eqref{fs} and then its inverse Fourier transform.

References

[1] Richard J. Briggs, Electron-stream Interaction with Plasmas, M.I.T. Press research monographs 29, Cambridge, Mass.: M.I.T. Press, pp. 187 (1964).

[2] Henrici, Peter, Applied and computational complex analysis. Vol. 2: Special functions-integral transforms-asymptotics-continued fractions, Wiley Classics Library. New York: Wiley. ix, 662 p. (1991). ZBL0925.30003.

[3] Tsai, C. C.; Young, D. L.; Fan, C. M.; Chen, C. W., "MFS with time-dependent fundamental solutions for unsteady Stokes equations", Engineering Analysis with Boundary Elements 30, No. 10, 897-908 (2006). ZBL1195.76324.