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Comptes Rendus
Mécanique
Ge Zu and Fang Li
Lower bound estimates of blow-up time for a quasilinear hyperbolic
equation with superlinear sources
Volume 348, issue 4 (2020), p. 307-313.
<https://doi.org/10.5802/crmeca.9>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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Comptes Rendus
Mécanique
2020, 348, n 4, p. 307-313
https://doi.org/10.5802/crmeca.9
Lower bound estimates of blow-up time for a
quasilinear hyperbolic equation with
superlinear sources
Ge Zua and Fang Li∗, a
a School of Mathematics, Jilin University, Changchun 130012, PR China
E-mails: zuge18@mails.jlu.edu.cn (G. Zu), lf8583@jlu.edu.cn (F. Li)
Abstract. This paper deals with the lower bound for blow-up solutions to a quasilinear hyperbolic equation
with strong damping. An inverse Hölder inequality with a correction constant is employed to overcome the
difficulty caused by the failure of the embedding inequality. Moreover, a lower bound for blow-up time is
obtained by constructing a new control functional with a small dissipative term and by applying an inverse
Hölder inequality as well as energy inequalities. This result gives a positive answer to the open problem
presented in [1].
Keywords. Inverse Hölder inequality, Energy estimate method, Energy inequality, Lower bound estimate,
Quasilinear hyperbolic equation.
Manuscript received 17th November 2018, revised 4th March 2019 and 14th September 2019, accepted 24th Feb-
ruary 2020.
1. Introduction
In this paper, the following quasilinear hyperbolic equation with strong damping is studied:
ut t −div(|∇u|p(x,t )−2∇u)−∆ut = |u|q(x,t )−2u, (x, t ) ∈Ω× (0,T ) :=QT
u(x, t ) = 0, (x, t ) ∈ ∂Ω× (0,T ) := ΓT
u(x,0) = u0(x), ut (x,0) = u1(x), x ∈Ω,
(1.1)
whereΩ⊂RN (N Ê 1) is a bounded domain with a smooth boundary ∂Ω, T > 0. It will be assumed
throughout this paper that the exponents p(x, t ) and q(x, t ) satisfy the following conditions:
2 É p− É p(x, t ) É p+ <∞, 1 < q− É q(x, t ) É q+ <∞.
Problem (1.1) models many physical problems such as viscoelastic fluids, electrorheological
fluids, processes of filtration through porous media, fluids with temperature-dependent viscosity,
and so on. The interested reader may refer to [2–4] and the references therein. In the case where
∗Corresponding author.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
308 Ge Zu et al.
p, q are fixed constants, many authors discussed the existence of solutions, finite-time blow-up
of solutions for low initial energy and arbitrarily high initial energy, and some estimate of a lower
bound for blow-up times. The interested reader may refer to [5–12]. In the case where p, q are
continuous functions, S. N. Antontsev [13, 14] studied the following problem:
ut t = div(a(x, t )|∇u|p(x,t )−2∇u)+α∆ut +b(x, t )|u|σ(x,t )−2u + f (x, t ), (x, t ) ∈Ω× (0,T )
u(x, t ) = 0, (x, t ) ∈ ∂Ω× (0,T )
u(x,0) = u0(x), ut (x,0) = u1(x), x ∈Ω.
(1.2)
Antontsev proved the existence and the blow-up of weak solutions for negative initial energy.
Later, Guo–Gao [15] discussed the blow-up properties of solutions to the above problems for the
case where the initial energy is positive. In addition, Messaoudi and Talahmeh [16, 17] discussed
blow-up properties of solutions to Problem (1.2) in the absence of a strong damping term.
It is well known that the source term causes finite-time blow-up of the solution while the
damping term may drive the equation toward stability. Therefore, it is of interest to explore
the mechanism of how sources dominate the dissipation (the damping term ∆ut ), which has
attracted considerable attention. In fact, the upper bound ensures the occurrence of blow-up
while the lower bound may provide us a safe time interval for operation when we use Problem
(1.1) to model a physical process. Hence, it is more interesting to give a lower bound estimate
for hyperbolic problems than to give a upper bound. In 2017, Guo [1] applied the modified
version of the Gagliardo–Nirenberg inequality for non-constant cases and energy inequalities
to obtain some estimates of lower bounds for blow-up time in the case where 2 < p− < q+ <
p−(1+ (2+p−∗)/2N ) with p−∗ = (N p−/(N −p−))(2 < p− < N ). In particular, Remark 1.1 of [1]
gives an unsolved problem, namely, as follows.
Remark 1.1. Since p ∈ [p−(1+(2+p−∗)/2N ), p−∗], it seems that we cannot obtain results similar
to those of Lemma 1.5 [1] unless we may obtain more information about ‖ut‖2. Therefore, we
need to develop a new method or technique to discuss this problem.
In this paper, we first follow along the lines of the proof of Lemma 1.3 [1] to obtain an inverse
Hölder inequality with correction constants in the case where p lies in [p−(1+(2+p−∗)/2N ), p−∗].
Second, we construct a new control functional with a small dissipative term and then apply
the inverse Hölder inequality as well as energy inequalities to establish a differential inequality.
Finally, we obtain an estimate of lower bounds for blow-up time.
This paper is organized as follows. First, in Section 2, we present some preliminaries. Section 3
is devoted to giving an estimation of a lower bound.
2. Preliminaries
Define the energy functional as
E(t ) = 1
2
∫
Ω
|ut |2 dx +
∫
Ω
1
p(x, t )
|∇u|p(x,t ) dx −
∫
Ω
1
q(x, t )
|u|q(x,t ) dx.
For simplicity, we give some notation and the embedding inequality to be used later. By Corollary
3.34 in [3], we know that W 1,p(x,0)0 (Ω) ,→W
1,p−
0 (Ω) ,→ Lr (Ω)(1 < r É (N p−/(N −p−))). Let B be the
best constant of the embedding inequality
‖u‖r É B‖∇u‖p(·), ∀ u ∈W 1,p(x,0)0 (Ω). (2.1)
Set E1 = (q+−p+)/q+p+α1,α1 = B (p
+q+)/(p+−q+)
1 , where B1 = max{B ,1}. The following conclu-
sions are presented to shorten the statement of our main results and their proofs.
C. R. Mécanique, 2020, 348, n 4, 307-313
Ge Zu et al. 309
Lemma 2.1 ( [15]). Suppose that u ∈ Lq(x,t )(QT )∩L∞(0,T ;W 1,p(x,t )0 (Ω)), and ut ∈ L2(0,T ; H 1(Ω))
is a solution to Problem (1.1). Then E(t ) satisfies the identity
E(t )+
∫ t
0
∫
Ω
|∇us |2 dx ds = E(0)+
∫ t
0
∫
Ω
ps (·)
p2(·) |∇u|
p(·) (ln |∇u|p(·) −1) dx ds
−
∫ t
0
∫
Ω
qs (·)
q2(·) |u|
q (ln |u|q −1)dx ds. (2.2)
Theorem 2.1 ( [15]). Assume that the initial data (u0,u1) and the exponents p(x, t ) and q(x, t )
satisfy the following conditions:
(H1) u0 ∈W 1,p(x,0)0 (Ω), u1 ∈ L2(Ω), E(0)+
|Ω|
p−
+ |Ω|
q−
< E1,
min
{
‖∇u0‖p
−
p(x,0),‖∇u0‖
p+
p(x,0)
}
>α1;
(H2) max{2, p
+} < q− É q(x, t ) É q+ < N p
−
N −p− , ∀ x ∈Ω, t Ê 0;
(H3) pt É 0, qt Ê 0,
∣∣∣∣ ptp2
∣∣∣∣+ ∣∣∣∣ qtq2
∣∣∣∣ ∈ L1l oc ((0,∞);L1(Ω)).
Then the solution to Problem (1.1) is not global.
Some ideas of this proof of Theorem 2.1 mainly come from the pioneering work of Levine [6,18]
(see also the work of Ball [19]). For more details, the reader may refer to [15].
Lemma 2.2 ( [15]). If u is the solution to Problem (1.1) and (H3) is satisfied, then the energy
functional E(t ) satisfies
E(t )+
∫ t
0
∫
Ω
|∇us |2 dx ds É E(0)+
(
1
p−
+ 1
q−
)
|Ω| := E2, t Ê 0. (2.3)
Lemma 2.3 ( [1]). Assume that u is the solution to Problem (1.1) and condition (H1) is fulfilled.
Then there exists a positive constant C depending on |Ω|, p−, N , and B1 such that for any k >
(N (q+−p−))/p−,∫
Ω
1
q(·) |u|
q(·) dx É 1
q−−p+ max{C
µ(k),Cν(k)}max
{(∫
Ω
|u|k dx
)α(k)
,
(∫
Ω
|u|k dx
)β(k)}
+ p
+
q−−p+
(
E2 + |Ω|
q−
)
. (2.4)
Here, µ, ν, α, and β are defined as follows:
µ(k) =

N (q+−k)
kp−−N (q+−p−) , k < q
+;
1− q
+
k
, k Ê q+.
ν(k) =

N p−(q+−k)
k(N p−−N p++p+p−)−N p−(q+−p+) , k < q
+;
1− q
+
k
, k Ê q+.
α(k) =

N p−−q+(N −p−)
kp−−N (q+−p−) , k < q
+;
q+
k
, k Ê q+.
C. R. Mécanique, 2020, 348, n 4, 307-313
310 Ge Zu et al.
β(k) =

[N p−−q+(N −p−)]p+
k(N p−−N p++p+p−)−N p−(q+−p+) , k < q
+;
q+
k
, k Ê q+.
In fact, when k < q+, we follow along the lines of the proof of Lemma 1.3 [1] to obtain the above
conclusions. When k Ê q+, we apply condition (H1) to prove that there exists a positive constant
α2 depending on E(0),B1 such that the term ‖u‖q(·) is bigger than α2. Then, we apply some
inequalities min{‖u‖q−q(·),‖u‖
q+
q(·)} É
∫
Ω |u|q(·) dx É max{‖u‖q
−
q(·),‖u‖
q+
q(·)} and ‖u‖q(·) É (1+|Ω|)‖u‖q+
to finish our proof.
3. Lower bound estimates
In this section, we give our main results and their proof.
Theorem 3.1. If (N (p−+2))/(2(N −p−)) < q+ < ((2N −p−+2)p−)/(2(N −p−)), then the blow-up
time T ∗ satisfies the estimate∫ +∞
F (0)
1
C4 y
2− 2
θ +C5 y2−
2
θ
+λ+C6 y2−
2
θ
+2λ dy ≤ T
∗.
Here, the constants C4 and C5 and the initial data F (0) are defined in (3.11) and
λ= N +2
2(N −p−) , θ =
2(N p−−q+N +q+p−)(N −p−)
(N +2)p−p−−2N (q+−p−)(N −p−) .
Proof. This proof is divided into three steps.
Step 1. Equivalent of blow-up. Define
H(t ) =
(∫
Ω
|u|k dx
)θ
− 1
2M
∫ t
0
∫
Ω
|∇uτ|2 dx dτ,
where
θ = 2(N p
−−q+N +q+p−)(N −p−)
(N +2)p−p−−2N (q+−p−)(N −p−) , M =
1
q−−p+ max{C
µ1 ,Cν1 }2θ−1,
µ1 = N (q
+−k)
kp−−N (q+−p−) , ν1 =
N p−(q+−k)
k(N p−−N p++p+p−)−N p−(q+−p+) , k =
N (p−+2)
2(N −p−) .
By Lemmas 2.2 and 2.3 and the definition of E(t ), we have∫ t
0
∫
Ω
|∇uτ|2 dx dτ≤ E2 +
∫
Ω
1
q(·) |u|
q(·) dx ≤ M
(∫
Ω
|u|k dx
)θ
+C1, (3.1)
where
C1 = M + p
+
q−−p+
(
E2 + |Ω|
q−
)
+E2.
The definition of H(t ) and Inequality (3.1) yield
H(t ) ≥ 1
2
(∫
Ω
|u|k dx
)θ
− C1
2M
. (3.2)
Combining the conclusion of Theorem 1.7 [1] with Inequality (3.2), we have
lim
t→T ∗
H(t ) =+∞. (3.3)
Step 2. A first-order differential inequality. A simple computation shows that
H ′(t ) = θk
(∫
Ω
|u|k dx
)θ−1 ∫
Ω
|u|(k−2)uut dx − 1
2M
∫
Ω
|∇ut |2 dx. (3.4)
C. R. Mécanique, 2020, 348, n 4, 307-313
Ge Zu et al. 311
By using the Hölder inequality, the Sobolev embedding theorem, and the Young inequality, it is
not hard to verify that
H ′(t ) ≤ θk
(∫
Ω
|u|k dx
)θ−1 (∫
Ω
|u|(k−1) 2NN+2 dx
) N+2
2N
(∫
Ω
|ut |2∗ dx
) 1
2∗ − 1
2M
∫
Ω
|∇ut |2 dx
≤ Cθk
(∫
Ω
|u|k dx
)θ−1 (∫
Ω
|u|(k−1) 2NN+2 dx
) N+2
2N
(∫
Ω
|∇ut |2 dx
) 1
2 − 1
2M
∫
Ω
|∇ut |2 dx
≤ MC
2
2
[
θk
(∫
Ω
|u|k dx
)θ−1 (∫
Ω
|u|(k−1) 2NN+2 dx
) N+2
2N
]2
+ 1
2M
∫
Ω
|∇ut |2 dx − 1
2M
∫
Ω
|∇ut |2 dx
≤ MC
2
2
[
θk
(∫
Ω
|u|k dx
)θ−1 (∫
Ω
|u|(k−1) 2NN+2 dx
) N+2
2N
]2
, (3.5)
where the constant C is the best embedding constant of the embedding H 10 (Ω) ,→ L(2N /N−2)(Ω).
In addition, noting that (2N (k −1))/(N +2) ≤ p−∗ and applying embedding inequality (2.1),
Lemmas 2.2 and 2.3, and the definition of E(t ), we have(∫
Ω
|u| 2N (k−1)N+2 dx
) N+2
2N ≤ B‖∇u‖k−1p(·) ≤ B max
{(∫
Ω
|∇u|p(·) dx
) k−1
p−
,
(∫
Ω
|∇u|p (·)dx
) k−1
p+
}
≤ B max
{(
p+E2 +
∫
Ω
p+
q(·) |u|
q(·) dx
) k−1
p−
,
(
p+E2 +
∫
Ω
p+
q(·) |u|
q(·) dx
) k−1
p+
}
≤ B max
{
1, M
k−1
p+ −
k−1
p−
1
}(
M1 +
∫
Ω
p+
q(·) |u|
q(·) dx
) k−1
p−
≤ B max
{
1, M
k−1
p+ −
k−1
p−
1
}(
M2 +p+M
(∫
Ω
|u|k dx
)θ) k−1p−
≤ C2 +C3
(∫
Ω
|u|k dx
) (k−1)θ
p−
, (3.6)
where the constants Ci (i = 2,3) are defined as follows:
C2 = 2
k−1
p−
(
p+E2 +2θp+M + p
+E2
q−−p+ +
p+|Ω|
(q−−p+)p−
) k−1
p−
B max
{
1,(p+E2)
k−1
p+ −
k−1
p−
}
,
C3 = B max
{
1,(p+E2)
k−1
p+ −
k−1
p−
}
(2θp+M)
k−1
p− .
Therefore, inserting (3.6) into (3.5), we get
H ′(t ) É MC
2θ2k2
2
(∫
Ω
|u|k dx
)2(θ−1) [
C2 +C3
(∫
Ω
|u|k dx
) (k−1)θ
p−
]2
. (3.7)
Step 3. A lower bound for blow-up time.
By using Inequality (3.2), (3.7) is equivalent to the inequality
H ′(t ) É MC
2θ2k2
2
(
2H(t )+ C1
M
)2(1− 1
θ
)
[
C2 +C3
(
2H(t )+ C1
M
) k−1
p−
]2
. (3.8)
Furthermore, a simple computation indicates that Inequality (3.8) may be rewritten as(
2H(t )+ C1
M
)′
É MC 2θ2k2
(
2H(t )+ C1
M
)2(1− 1
θ
)
[
C2 +C3
(
2H(t )+ C1
M
) k−1
p−
]2
. (3.9)
C. R. Mécanique, 2020, 348, n 4, 307-313
312 Ge Zu et al.
Setting F (t ) = 2H(t )+C1/M , we have
F ′(t ) É MC 2θ2k2F 2(1− 1θ )(t )
[
C2 +C3F
k−1
p− (t )
]2
:= C4F 2(1−
1
θ
)(t )+C5F 2(1−
1
θ
)+ k−1p− (t )+C6F 2(1−
1
θ
)+ 2(k−1)p− (t ), (3.10)
where
C4 = MC 2k2θ2C 22 , C5 = 2MC 2θ2k2C2C3,
C6 = MC 2θ2k2C 23 , F (0) = 2
(∫
Ω
|u0|k dx
)θ
+ C1
M
.
(3.11)
Equation (3.10) implies∫ +∞
F (0)
1
C4 y
2− 2
θ +C5 y2−
2
θ
+ k−1p− +C6 y2−
2
θ
+ 2(k−1)p−
dy ≤ T ∗.
This completes the proof of this theorem. 
Remark 3.1. The fact
(2N −p−+2)p−
2(N −p−) −p
−
(
1+ 2+p
−∗
2N
)
= p
−p−
N (N −p−) > 0
shows that the result of this paper gives a positive answer to the unsolved problem in [1]. However,
when q+ lies in the interval [((2N −p−+2)p−)/(2(N −p−)), p−∗], due to technical reasons, at
present, we cannot give any answer.
Acknowledgment
This research was supported by the Scientific and Technological Project of Department of Edu-
cation of Jilin Province in Thirteenth-five-Year (JJKH20180111KJ).
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C. R. Mécanique, 2020, 348, n 4, 307-313


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Patrick Huerre
Yves Couder (1941–2019): A life in search of the beauty of fluid motion
Volume 348, issue 6-7 (2020), p. 397-400.
<https://doi.org/10.5802/crmeca.37>
Part of the Thematic Issue: Tribute to an exemplary man: Yves Couder
Guest editors:Martine Ben Amar (Paris Sciences & Lettres, LPENS, Paris, France),
Laurent Limat (Paris-Diderot University, CNRS, MSC, Paris, France),
Olivier Pouliquen (Aix-Marseille Université, CNRS, IUSTI, Marseille, France)
and Emmanuel Villermaux (Aix-Marseille Université, CNRS, Centrale Marseille,
IRPHE, Marseille, France)
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Comptes Rendus
Mécanique
2020, 348, n 6-7, p. 397-400
https://doi.org/10.5802/crmeca.37
Tribute to an exemplary man: Yves Couder
Hommage to the man and his career
Yves Couder (1941–2019): A life in search of
the beauty of fluid motion
Patrick Huerrea
a Laboratoire d’Hydrodynamique - LadHyX, CNRS – Ecole Polytechnique, Institut
Polytechnique de Paris, F – 91128 Palaiseau, France
E-mail: huerre@ladhyx.polytechnique.fr
Abstract. In this article, I briefly discuss the remarkable scientific accomplishments of Yves Couder, their
impact in the fluid dynamics community worldwide and his unique personality. It is a personal tribute to a
friend and a colleague I have known for the past 35 years.
Keywords. Soap films, Turbulence, Fingering, Morphogenesis, Drop dynamics.
Yves Couder passed away at age 78 on April 2, 2019 in Paris, surrounded by his wife Lorna and
their two children Jeanne and Julien. Many of his friends and colleagues attended the memorial
and burial ceremonies held on April 8, 2019.
He was Professor of Physics at the Denis Diderot University in Paris and a member of the
Académie des Sciences.
After completing his doctorate in physics at the University of Paris in 1965, Yves started
working at Ecole normale supérieure where he created and developed a talented and thriving
research team. In 1985, he also became Professor of Physics at the Paris Denis Diderot University
where, as of 2006, his research activities moved within the Matière et Systèmes Complexes (Matter
and Complex Systems) laboratory. He played a pivotal role in the foundation and establishment
of this new lab.
Yves Couder’s research achievements in Fluid Dynamics are simply stellar: he has made pace-
setting and truly seminal contributions in several important areas of the field by conceiving
deceptively simple experiments. Here are some of his most striking accomplishments in Fluid
Physics and in closely connected areas.
Soap films, dynamics of vortices and two-dimensional turbulence. Yves Couder and colleagues
[1,2] were the first to introduce soap films as an original setting to study the dynamics of vortices
and the properties of two-dimensional turbulence. In particular he was able to observe and char-
acterize the inverse energy cascade predicted by the theory of two-dimensional turbulence. This
very productive analogy subsequently led to the development of other soap film experiments in
several groups in the USA and Europe.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
398 Patrick Huerre
Figure 1. Yves Couder and Stéphane Perrard, 2011.
Three-dimensional turbulence and vortex filaments. Yves Couder and colleagues [3] were also
the first to detect and characterize intense vortex filaments in three-dimensional turbulence.
They showed that such filaments, referred to as the sinews of turbulence by Keith Moffatt, are
responsible for the production of low-pressure bursts in the so-called French washing machine
experiment.
Saffman-Taylor fingering. This phenomenon refers to the formation of fingers at the interface
between two fluids when a more viscous fluid is displaced by another less viscous fluid. Yves [4]
demonstrated that the application of singular perturbations at the tip of the finger is sufficient to
influence the selection process that determines the finger configuration.
Ferromagnetic drops, morphogenesis and the growth of plants. The spiral patterns displayed
by plants may be explained by resorting to the properties of Fibonacci sequences and the Golden
Mean. Yves has confirmed these features by studying in detail the spiral patterns formed by the
arrangement of seeds in the central bud of sunflowers [5, 6]. Here again he proposed a striking
analogy between this process and pattern formation in a collection of ferromagnetic drops. This
fluid experiment led to a global interpretation of the observed patterns: the system of drops tends
to a configuration based on irrational numbers since, during the pattern formation process, it is
observed to avoid rational numbers.
Particle-wave duality in drop dynamics. In his last years (Figures 1 and 2), Yves Couder was en-
gaged in a comprehensive investigation of the dynamics of drops that bounce on a liquid inter-
face subjected to periodic vertical oscillations (Faraday experiment). This set-up revealed the ex-
istence of a dynamic coupling between the vertical motion of the bouncing drop and the gravity
C. R. Mécanique, 2020, 348, n 6-7, 397-400
Patrick Huerre 399
Figure 2. Yves Couder and Emmanuel Fort, 2011.
wave field which it generates [7, 8]. The study of this phenomenon has led to fascinating macro-
scopic analogs of G.I. Taylor’s single-photon experiment and de Broglie’s electron diffraction ex-
periment [9]. Chaotic régimes were more recently identified [10] and Yves’ pioneering studies
have triggered further important research in the USA.
John Bush, Professeur of Mathematics at MIT, whose recent work has been inspired by
Couder’s, states: “In demonstrating the possibility of both observing and understanding this phe-
nomenon on a macroscopic scale, Couder’s experiments invite us to reconsider the philosoph-
ical foundations of quantum mechanics. What better role for fluid mechanics than questioning
the foundations of modern physics? What better goal than the prospect of restoring determinism
and realism to science ? Indeed it is (one’s) hope that hydrodynamic quantum analogue systems,
as initiated by (Yves) Couder, will establish fluid mechanics as a subject worthy of representa-
tion in modern physics departments worldwide. Couder’s research (presently) continues in the
pioneering spirit of G.I. Taylor” (John Bush private communication).
During his career, Yves received many distinctions: the Prize of Institut Français du Pétrole
of the Académie des Sciences in 1993, the Gentner-Kastler Prize jointly awarded by the Deutsche
Physikalische Gesellschaft and the Société Française de Physique in 2006, the Trois Physiciens
(Abraham, Bloch et Bruhat) Prize in 2007. He received in 2012 the EUROMECH Fluid Mechan-
ics Prize of the European Mechanics Society with the simple, yet very meaningful citation: for ex-
periments in fluid mechanics which are novel, elegant, deep and provocative.
Yves’ creativity was exceptional and he had a special talent for detecting profound analogies
between seemingly very distinct fields. He has had a profound impact on colleagues and young
brilliant researchers both in the US and in Europe, who constitute what may rightfully be called
an informal Couder school of macroscopic physics.
As the reader of these lines may already know, Yves was the kind of Man one rarely happens to
meet during a researcher’s career. As far as I am concerned, it was on the occasion of my visit at
Ecole Normale Supérieure in Paris in 1985 that I had the chance of getting to know him. In those
years, he worked with Marc Rabaud and Jean-Marc Chomaz on obtaining strikingly beautiful
images of the dynamics of vortices in liquid soap films (see above).
C. R. Mécanique, 2020, 348, n 6-7, 397-400
400 Patrick Huerre
One could not avoid being seduced by the disconcerting, yet deep, freshness of his intellect. A
few weeks before his passing, as he was severely weakened, I was once more surprised by his cor-
rosive intuition regarding the different types of memory, when he juggled from his experiments
on jumping drops (see above) to rebound on his thoughts relative to the distinct forms of memory
of the human mind.
This was also obvious in the way he interacted with friends and colleagues. I will never forget
his warm voice and the discretely amused twinkles in his eye. When one came to know him better,
one would discover a genuine modesty. No incongruous pomposity in communicating the results
of his own research but rather an immediate opening to address the next relevant and pressing
questions! The new developments, although remarkable, would almost be relegated in favour of
really new unresolved issues!
Thus, it is on many grounds that Yves was elected to the Académie des Sciences in 2013
within the section of Mechanical and Computational Sciences. We both used to joke about the
coexistence of two distinct communities of researchers in fluid dynamics: physicists and for lack
of an equivalent word in English, mécaniciens. These two varieties of fluid dynamicists were
frequently rivals but they were unanimous in welcoming Yves within the Academy. During our
debates regarding the selection of candidates for various prizes, he would focus his attention
on the major scientific achievements of the candidates and bypass most biographical aspects in
their career, which to him, appeared to be anecdotal. Once again, this was typical of his approach,
away from the marked trails and right to the essentials.
Last but not least, on a more personal note, I fondly keep in my mind the lively and lovely
dinners at Lorna and Yves’ home on La butte aux cailles in Paris where one would always have
stimulating conversations in a convivial atmosphere.
Yves Couder is sorely missed in our community but for years to come, he will remain a vibrant
source of inspiration. . .
Acknowledgements
The author is grateful to Julien Moukhtar and to Laurent Limat, director of the Matière et Systèmes
Complexes laboratory for providing the two photographs of Yves Couder which are included in
the present article.
References
[1] Y. Couder, C. Basdevant, “Experimental and numerical study of vortex couples on two-dimensional flows”, J. Fluid
Mech. 173 (1986), p. 225-251.
[2] Y. Couder, J. M. Chomaz, M. Rabaud, “On the hydrodynamics of soap films”, Physica D 37 (1989), p. 384-405.
[3] S. Douady, Y. Couder, M. E. Brachet, “The direct observation of the intermittency of intense vorticity filaments in
turbulence”, Phys. Rev. Lett. 67 (1991), p. 983-986.
[4] Y. Couder, N. Gérard, M. Rabaud, “Narrow fingers in the Saffman–Taylor instability”, Phys. Rev. Lett. 34 (1986),
p. 5175-5178.
[5] S. Douady, Y. Couder, “Phyllotaxis as a physical self-organized growth process”, Phys. Rev. Lett. 68 (1992), p. 2098-
2101.
[6] S. Douady, Y. Couder, “Phyllotaxis as a dynamical self-organizing process. 1. The spiral modes resulting from time-
periodic iterations”, J. Theoret. Biol. 178 (1996), p. 255-274.
[7] Y. Couder, S. Protière, E. Fort, A. Boudaoud, “Dynamical phenomena – Walking and orbiting droplets”, Nature 437
(2005), no. 7056, p. 208.
[8] S. Protière, A. Boudaoud, Y. Couder, “Particle-wave association on a fluid interface”, J. Fluid Mech. 554 (2006), p. 85-
108.
[9] Y. Couder, E. Fort, “Single particle diffraction and interference at a macroscopic scale”, Phys. Rev. Lett. 97 (2006),
154101.
[10] S. Perrard, M. Labousse, M. Miskin, E. Fort, Y. Couder, “Self-organization into quantized eigenstates of a classical
wave-driven particle”, Nat. Commun. 5 (2014), p. 3219.
C. R. Mécanique, 2020, 348, n 6-7, 397-400


Tribute to an exemplary man: Yves Couder

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