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Long-time behavior of several point particles in a 1D viscous compressible fluid

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Abstract

We study the long-time behavior of several point particles in a 1D viscous compressible fluid. It is shown that the velocities of the point particles all obey the power law \(t^{-3/2}\). This result extends author’s previous works on the long-time behavior of a single point particle. New difficulties arise in the derivation of pointwise estimates of Green’s functions due to infinite reflections of waves in-between the point particles. In particular, the differential equation technique used in previous works alone does not suffice. We overcome this by carefully analyzing the structure of Green’s functions in the Laplace variable, especially their asymptotic and analyticity properties.

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Notes

  1. Some authors prefer to call it a piston. This is just a matter of taste. We chose the terminology a point particle to emphasise that the problem is considered in a one-dimensional setting.

  2. Similar decay estimates for \(\rho -1\) and U can also be obtained by noting that the change of variable from the Lagrangian mass coordinate x to the Eulerian coordinate X satisfies \(\partial X/\partial x=v\) and that v is bounded from above and below by positive constants uniformly in time when \(\delta '\) is sufficiently small.

  3. The interpretation below is inspired by a closely related analysis in [18] for a heat equation with a conductivity having jumps.

  4. We only considered the case of \(x>0\) above but the case of \(x<0\) is similar.

  5. More precisely, it might contain a Dirac delta singularity but \(H-G\) should be a usual function.

  6. Directly applying Proposition 3.2 for the derivatives of G results in apparently diverging integrals. We can circumvent this problem by noting that applying \(\partial _{x}^{k}\) is equivalent to multiplying \((-\lambda )^k\) in the Laplace transformed side, and the divergence of \(\lambda ^k\) as \(|s|\rightarrow \infty \) is then absorbed by the exponential factor in \({\mathcal {L}}[\omega ](s)\).

  7. This is a simplified explanation and some technical arguments are needed to make it rigorous; see [14, Sect. 3.3].

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Acknowledgements

I thank Shih-Hsien Yu for informing me about some ideas in [18] (which at the time was an unpublished preprint) when I visited National University of Singapore in 2019, which was financially supported by Grant-in-Aid for JSPS Research Fellow (Grant Number 18J20574). This motivated me to consider the problem presented in this paper, and Proposition 3.1 emerged as an application of one of the ideas in their paper. This work was financially supported by Grant-in-Aid for JSPS Research Fellow (Grant Number 20J00882).

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  1. Department of Mathematics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

    Kai Koike

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A lemma on the quantity \(\lambda =s/\sqrt{\nu s+c^2}\)

A lemma on the quantity \(\lambda =s/\sqrt{\nu s+c^2}\)

Lemma A.1

Let \(\lambda =s/\sqrt{\nu s+c^2}\) and \(r=\lambda ^2/(\lambda +2)^2\). Then the following properties are satisfied:

  1. (i)

    There exist \(\sigma _0>0\) and \(r_0 \in (0,1)\) such that \({\text {Re}}\lambda \ge -r_0\) and \(|re^{-2\lambda }|\le r_0\) for all \(s\in {\mathbb {C}}\backslash (-\infty ,-c^2/\nu ]\) with \({\text {Re}}s\ge -\sigma _0\).

  2. (ii)

    We have

    $$\begin{aligned} {\text {Re}}\lambda \ge \frac{\sqrt{|s|}}{2\sqrt{\nu }}+O\left( \frac{1}{\sqrt{|s|}} \right) \quad (|s|\rightarrow \infty ; {\text {Re}}s>-c^2/\nu ). \end{aligned}$$

Proof

Note first that \({\text {Re}}\lambda \ge 0\) for \({\text {Re}}s\ge 0\). From this, it follows that for any \(M>0\), there exists \(r_0 \in (0,1)\) such that \(|re^{-2\lambda }|\le r_0\) for all s with \({\text {Re}}s\ge 0\) and \(|s|\le M\). Then, if (ii) is proved, (i) follows easily.

Now we prove (ii). Let s be a sufficiently large complex number with \({\text {Re}}s>-c^2/\nu \). We can write it as \(s=|s|e^{i\theta }\) with \(\theta \in [-2\pi /3,2\pi /3]\). Then, from

$$\begin{aligned} \lambda =\frac{s}{\sqrt{\nu s}}\frac{\sqrt{\nu s}}{\sqrt{\nu s+c^2}}=\frac{\sqrt{s}}{\sqrt{\nu }}\left[ 1+O\left( \frac{1}{s} \right) \right] =\frac{\sqrt{|s|}}{\sqrt{\nu }}e^{\frac{i\theta }{2}}\left[ 1+O\left( \frac{1}{s} \right) \right] , \end{aligned}$$

we obtain

$$\begin{aligned} {\text {Re}}\lambda =\frac{\sqrt{|s|}}{\sqrt{\nu }}\cos \left( \frac{\theta }{2} \right) +O\left( \frac{1}{\sqrt{|s|}} \right) \ge \frac{\sqrt{|s|}}{2\sqrt{\nu }}+O\left( \frac{1}{\sqrt{|s|}} \right) . \end{aligned}$$

This ends the proof of the lemma. \(\square \)

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Koike, K. Long-time behavior of several point particles in a 1D viscous compressible fluid. J. Evol. Equ. 22, 61 (2022). https://doi.org/10.1007/s00028-022-00820-8

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