Comptes Rendus Mathématique

Pierre Deligne

Comptes Rendus
Mathématique
Pierre Deligne
Le critère d’Abel pour la résolubilité par radicaux d’une équation
irréductible de degré premier
Volume 359, issue 7 (2021), p. 919-921
<https://doi.org/10.5802/crmath.242>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 7, p. 919-921
https://doi.org/10.5802/crmath.242
Théorie des Groupes / Group theory
Le critère d’Abel pour la résolubilité par
radicaux d’une équation irréductible de degré
premier
Abel’s criterion for the solvability by radicals of an
irreducible equation of prime degree
Pierre Deligne∗, a
a Institute for Advanced Study, Princeton, NJ 08540, USA.
Courriel : deligne@math.ias.edu
Résumé. Dans sa dernière lettre à Crelle, Abel énonce un critère pour déterminer si une équation irréductible
de degré premier est résoluble par radicaux. Sylow dit cet énoncé ambigu et devant être modifié. Nous
montrons que le critère donné par Abel est correct tel quel.
Abstract. In his last letter to Crelle, Abel states a criterion for the solvability by radicals of an irreducible
equation of prime degree. Sylow finds Abel’s statement ambiguous, and writes that it should be modified.
We show the correctness of Abel’s original statement.
Manuscrit reçu le 21 août 2020, révisé le 1 juin 2021, accepté le 5 juillet 2021.
Dans sa dernière lettre à Crelle, datée du 18 Octobre 1828 et reproduite dans [1], Abel énonce
le critère suivant pour qu’une équation irréductible de degré premier soit résoluble par radicaux :
« Si trois racines d’une équation quelconque irréductible d’un degré marqué par
un nombre premier, sont liées entre elles de la manière que l’on pourra exprimer
l’une de ces racines rationnellement en les deux autres, l’équation sera toujours
résoluble par radicaux. »
La théorie de Galois fournit du critère d’Abel la traduction suivante.
Théorème. Soient p un nombre premier et G un groupe transitif de permutations d’un ensemble
E à p éléments. Pour que le groupe G soit résoluble il (faut et il) suffit que quel que soit le choix de
trois éléments dans E, il existe un de ces trois éléments qui est fixé par tous les éléments de G qui
fixent les deux autres.
∗Auteur correspondant.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
920 Pierre Deligne
Dans l’article [3] à l’occasion du centenaire de la naissance d’Abel, Sylow écrit page 18 :
« Abel s’est toutefois exprimé inexactement ou au moins peu clairement, et
a sans doute voulu dire que toutes les racines devaient pouvoir s’exprimer
rationnellement en fonction de deux d’entre elles, toujours les mêmes. Prise
rigoureusement à la lettre, la proposition est inexacte, car il existe une classe
d’équations à laquelle elle ne s’applique pas, tout ou moins lorsque le degré est
de la forme 2k −1. »
Nous montrons que, nonobstant ce que dit Sylow, l’énoncé d’Abel est correct. Nous n’avons
toutefois aucune idée de quelle était sa preuve.
La preuve du lemme qui suit est due à Galois [2, propositions VI, VII et VIII].
Lemme. Avec les hypothèses et notations du théorème, les conditions suivantes sont équivalentes :
(i) Il existe deux éléments de E tels que le seul élément de G qui les fixe soit l’identité.
(ii) |G| < p2.
(iii) G contient un sous-groupe distingué A d’ordre p.
(iv) Une indexation Z/p →̃ E transforme G en un sous-groupe du groupe des affinités x 7→
ax +b, contenant les translations.
(v) G est résoluble.
Démonstration.
(i) ⇒ (ii). En effet, (i) implique que |G| ≤ p (p −1), le nombre de paires ordonnées d’éléments
de E .
(ii)⇒ (iii). Puisque G agit transitivement sur E , p divise l’ordre |G| de G et, d’après un théorème
de Cauchy, G contient un sous-groupe cyclique A d’ordre p. Il ne peut en contenir un deuxième,
B , car on aurait dans ce cas |AB | = p2 ≤ |G|. Le sous-groupe A, étant unique, est distingué.
(iii) ⇒ (iv). Identifions E à Z/p de sorte que A soit le groupe des translations. Le groupe G est
alors contenu dans le groupe affine des x 7→ ax +b, qui est le normalisateur de A dans le groupe
symétrique SE .
(iv) ⇒ (i) et (v). Clair.
(v) ⇒ (iv). Soit {e} = G0 ⊂ G1 ⊂ . . . ⊂ Gn = G une suite strictement croissante de sous-groupes
de G , avec Gi distingué dans Gi+1 et Gi+1/Gi cyclique. Par hypothèse, l’action de G sur E est
transitive et fidèle. Si H ⊂ G est un sous-groupe distingué, ses orbites dans E ont toutes le même
nombre d’éléments. Si H 6= {e}, H est donc transitif. Appliquant cet argument aux Gi , on voit
par récurrence descendante que les Gi (i > 0) sont transitifs. Le groupe G1 est commutatif et agit
fidèlement. Il est donc cyclique d’ordre p. Comme dans la preuve de (iii) ⇒ (iv), on peut identifier
E à Z/p de sorte que G1 soit le groupe des translations. Puisque G2 normalise G1, il est contenu
dans le groupe affine, et G1 est l’unique sous-groupe d’ordre p de G2. Puisque G2 est distingué
dans G3, G3 normalise le sous-groupe caractéristique G1 de G2, et est contenu dans le groupe
affine. Répétant l’argument jusqu′à atteindre Gn , on obtient (iv). 
Preuve du théorème. Si G est résoluble, on applique la partie (v) ⇒ (iv) du lemme. Réciproque-
ment, supposons que G vérifie la condition énoncée par Abel. Nous prouverons (i) du lemme.
Puisque G agit transitivement sur E , p = |E | divise l’ordre de G , et G contient une permutation
h de E d’ordre p. Identifions E à Z/p, de sorte que h soit la permutation n 7→ n +1 de Z/p. Pour
vérifier (i) du lemme, nous n’aurons à utiliser la condition énoncée par Abel que pour les triplets
{a,b,c} en progression arithmétique.
Pour p = 2, l’énoncé est trivial. Supposons p > 2. Supposons d’abord que pour un k 6= 0 et un
triplet de la forme {a, a +k, a +2k}, tout g dans G fixant a et a +k fixe aussi a +2k. Conjuguant
par une puissance de h, on obtient le même énoncé pour tout a. Si g fixe a et a +k, il fixe (a +k)
C. R. Mathématique — 2021, 359, n 7, 919-921
Pierre Deligne 921
et (a +k)+k, et donc a +3k. Itérant l’argument, on voit que g fixe tous les a +nk, c’est à dire E
tout entier. La clause (i) du lemme est donc vérifiée par a et a +k.
Appliquant ceci à - k et au triplet {a +2k, a +k, a}, on voit que si tout g dans G fixant a +2k et
a +k fixe aussi a, la clause (i) du lemme est vérifiée par a +2k et a +k.
Si pour aucun triplet en progression arithmétique ces arguments ne permettent de conclure,
alors, quels que soient a et b distincts, si g dans G fixe a et b, g fixe aussi 12 (a +b). Sous cette
hypothèse, montrons par récurrence sur n que si g fixe 0 et b, g fixe aussi les 2−n .i .b pour
0 ≤ i ≤ 2n . Considérant deux i consécutifs, et leur moyenne, on obtient en effet les 2−(n+1)i .b
(i impair) à partir des 2−n .i .b. Dès que 2n ≥ p, E entier est atteint de sorte que la clause (i) du
lemme est vérifiée par 0 et b. 
Références
[1] N. H. Abel, Oeuvres complètes. Nouvelle édition publiée aux frais de l’État Norvégien par MM. L. Sylow et S. Lie.,
Christiania. Grøndahl et Søn, 1881, Tome 2, XXII, p. 270.
[2] É. Galois, « Mémoire sur les conditions de résolubilité des équations par radicaux », J. Math. Pures et Appl. 11 (1846),
p. 417-433.
[3] L. Sylow, « Les études d’Abel et ses découvertes », in Niels Henrik Abel, Mémorial publié à l’occasion du centenaire de
sa naissance, Christiania. Grøndahl et Søn, 1902.
C. R. Mathématique — 2021, 359, n 7, 919-921


Le critère d’Abel pour la résolubilité par radicaux d’une équation irréductible de degré premier

Kanishka Perera

Comptes Rendus
Mathématique
Kanishka Perera
On the existence of ground state solutions to critical growth problems
nonresonant at zero
Volume 359, issue 9 (2021), p. 1161-1164
<https://doi.org/10.5802/crmath.270>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 9, p. 1161-1164
https://doi.org/10.5802/crmath.270
Partial differential equations / Equations aux dérivées partielles
On the existence of ground state solutions to
critical growth problems nonresonant at zero
Kanishka Pereraa
a Department of Mathematical Sciences, Florida Institute of Technology, Melbourne,
FL 32901, USA
E-mail: kperera@fit.edu
Abstract. We prove the existence of ground state solutions to critical growth p-Laplacian and fractional p-
Laplacian problems that are nonresonant at zero.
2020 Mathematics Subject Classification. 35B33, 35J92, 35R11.
Manuscript received 23rd June 2021, accepted 8th September 2021.
Consider the problem {
−∆p u =λ |u|p−2 u +|u|p
∗−2 u inΩ
u = 0 on ∂Ω, (1)
where Ω is a bounded domain in RN , 1 < p < N , ∆p u = div(|∇u|p−2∇u) is the p-Laplacian of u,
λ ∈ R, and p∗ = N p/(N −p) is the critical Sobolev exponent. Solutions of this problem coincide
with critical points of the C 1-functional
E(u) = 1
p
∫
Ω
|∇u|p dx − λ
p
∫
Ω
|u|p dx − 1
p∗
∫
Ω
|u|p∗dx, u ∈W 1,p0 (Ω).
Let K = {u ∈W 1,p0 (Ω) \ {0} : E ′(u) = 0} be the set of nontrivial critical points of E and set
c = inf
u∈K
E(u).
Recall that u0 ∈ K is called a ground state solution if E(u0) = c. For each u ∈ K ,
E(u) = E(u)− 1
p∗
E ′(u)u = 1
N
∫
Ω
|u|p∗dx > 0,
so c ≥ 0, and c > 0 if there is a ground state solution. Let
S = inf
u∈D1,p (RN )\{0}
∫
RN
|∇u|p dx(∫
RN
|u|p∗dx
)p/p∗
be the best Sobolev constant. Denote byσ(−∆p ) the Dirichlet spectrum of −∆p inΩ consisting of
those λ ∈R for which the eigenvalue problem{−∆p u =λ |u|p−2 u inΩ
u = 0 on ∂Ω (2)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1162 Kanishka Perera
has a nontrivial solution. We have the following theorem.
Theorem 1. If problem (1) has a nontrivial solution u with
E(u) < 1
N
SN /p (3)
and λ ∉σ(−∆p ), then it has a ground state solution.
Proof. Let
(
u j
) ⊂ K be a minimizing sequence for c. Then (u j ) is a (PS)c sequence for E . Since
problem (1) has a nontrivial solution satisfying (3), c < SN /p /N . So E satisfies the (PS)c condition
(see Guedda and Véron [6, Theorem 3.4]). Hence a renamed subsequence of
(
u j
)
converges to
a critical point u0 of E with E(u0) = c. We claim that u0 is nontrivial and hence a ground state
solution of problem (1). To see this, suppose u0 = 0. Then ρ j := ‖u j ‖ → 0. Let ũ j = u j /ρ j . Since
‖ũ j ‖ = 1, a renamed subsequence of
(
ũ j
)
converges to some ũ weakly in W 1,p0 (Ω), strongly in
Lp (Ω), and a.e. inΩ. Since E ′(u j ) = 0,∫
Ω
|∇u j |p−2∇u j ·∇v dx =λ
∫
Ω
|u j |p−2 u j v dx +
∫
Ω
|u j |p
∗−2 u j v dx ∀ v ∈W 1,p0 (Ω),
and dividing this by ρp−1j gives∫
Ω
|∇ũ j |p−2∇ũ j ·∇v dx =λ
∫
Ω
|ũ j |p−2 ũ j v dx +o(‖v‖) ∀ v ∈W 1,p0 (Ω). (4)
Passing to the limit in (4) gives∫
Ω
|∇ũ|p−2∇ũ ·∇v dx =λ
∫
Ω
|ũ|p−2 ũv dx ∀ v ∈W 1,p0 (Ω),
so ũ is a weak solution of (2). Taking v = ũ j in (4) and passing to the limit shows that
λ
∫
Ω |ũ|p dx = 1, so ũ is nontrivial. This contradicts the assumption that λ ∉ σ(−∆p ) and com-
pletes the proof. 
Combining this theorem with the existence results in García Azorero and Peral Alonso [5],
Egnell [4], Guedda and Véron [6], Arioli and Gazzola [1], and Degiovanni and Lancelotti [3] gives
us the following theorem for the case N ≥ p2.
Theorem 2. If N ≥ p2 and λ ∈ (0,∞) \σ(−∆p ), then problem (1) has a ground state solution.
For N < p2, combining Theorem 1 with Perera et al. [10, Corollary 1.2] gives the following
theorem, where (λk ) ⊂ σ(−∆p ) is the sequence of eigenvalues based on the Z2-cohomological
index introduced in Perera [8] and | · | denotes the Lebesgue measure in RN .
Theorem 3. If N < p2 and
λ ∈
∞⋃
k=1
(
λk −
S
|Ω|p/N ,λk
)
\σ(−∆p ),
then problem (1) has a ground state solution.
Remark 4. In the semilinear case p = 2, Theorem 2 was proved in Szulkin et al. [11] using a
Nehari–Pankov manifold approach, and Theorems 1 and 3 were proved in Chen et al. [2] using
a more direct approach. Moreover, they allow λ to be an eigenvalue when N ≥ 5. However, their
proofs are strongly dependent on the fact that H 10 (Ω) splits into the direct sum of its subspaces
spanned by the eigenfunctions of the Laplacian that correspond to eigenvalues that are less than
or equal to λ and those that are greater than λ. Those proofs do not extend to the p-Laplacian
since it is a nonlinear operator and hence has no linear eigenspaces.
Remark 5. We conjecture that the assumption λ ∉ σ(−∆p ) can be removed from Theorems 1
and 2 when N 2/(N +1) > p2.
C. R. Mathématique — 2021, 359, n 9, 1161-1164
Kanishka Perera 1163
Our argument can be easily adapted to obtain ground state solutions of other types of critical
growth problems as well. For example, consider the nonlocal problem{
(−∆)sp u =λ |u|p−2 u +|u|p
∗
s −2 u inΩ
u = 0 in RN \Ω, (5)
whereΩ is a bounded domain in RN with Lipschitz boundary, s ∈ (0,1), 1 < p < N /s, (−∆)sp is the
fractional p-Laplacian operator defined on smooth functions by
(−∆)sp u(x) = 2 lim
ε↘0
∫
RN \Bε(x)
|u(x)−u(y)|p−2 (u(x)−u(y))
|x − y |N+sp dy, x ∈R
N ,
λ ∈R, and p∗s = N p/(N − sp) is the fractional critical Sobolev exponent. Let | · |p denote the norm
in Lp (RN ), let
[u]s,p =
(∫
R2N
|u(x)−u(y)|p
|x − y |N+sp dxdy
)1/p
be the Gagliardo seminorm of a measurable function u :RN →R, and let
W s,p (RN ) = {u ∈ Lp (RN ) : [u]s,p <∞}
be the fractional Sobolev space endowed with the norm
‖u‖s,p =
(|u|pp + [u]ps,p)1/p .
We work in the closed linear subspace
W s,p0 (Ω) =
{
u ∈W s,p (RN ) : u = 0 a.e. in RN \Ω}
equivalently renormed by setting ‖·‖ = [ · ]s,p . Solutions of problem (5) coincide with critical
points of the C 1-functional
Es (u) = 1
p
∫
R2N
|u(x)−u(y)|p
|x − y |N+sp dxdy −
λ
p
∫
Ω
|u|p dx − 1
p∗s
∫
Ω
|u|p∗s dx, u ∈W s,p0 (Ω).
As before, a ground state is a least energy nontrivial solution. Let
Ẇ s,p (RN ) =
{
u ∈ Lp∗s (RN ) : [u]s,p <∞
}
endowed with the norm ‖·‖ and let
S = inf
u∈Ẇ s,p (RN )\{0}
∫
R2N
|u(x)−u(y)|p
|x − y |N+sp dxdy(∫
RN
|u|p∗s dx
)p/p∗s
be the best fractional Sobolev constant. Denote by σ((−∆)sp ) the Dirichlet spectrum of (−∆)sp in
Ω consisting of those λ ∈R for which the eigenvalue problem{
(−∆)sp u =λ |u|p−2 u inΩ
u = 0 in RN \Ω
has a nontrivial solution. Following theorem can be proved arguing as in the proof of Theorem 1.
Theorem 6. If problem (5) has a nontrivial solution u with
Es (u) < s
N
SN /sp
and λ ∉σ((−∆)sp ), then it has a ground state solution.
Combining this theorem with the existence results in Mosconi et al. [7] and Perera et al. [9]
gives us the following theorem, where (λk ) ⊂ σ((−∆)sp ) is the sequence of eigenvalues based on
the Z2-cohomological index.
C. R. Mathématique — 2021, 359, n 9, 1161-1164
1164 Kanishka Perera
Theorem 7. Problem (5) has a ground state solution in each of the following cases:
(i) N > sp2 and λ ∈ (0,∞) \σ((−∆)sp ),
(ii) N = sp2 and λ ∈ (0,λ1),
(iii) N ≤ sp2 and
λ ∈
∞⋃
k=1
(
λk −
S
|Ω|sp/N ,λk
)
\σ((−∆)sp ).
Remark 8. Theorems 6 and 7 are new even in the semilinear case p = 2.
Remark 9. We conjecture that problem (5) has a ground state solution for all λ > 0 when
N 2/(N + s) > sp2.
References
[1] G. Arioli, F. Gazzola, “Some results on p-Laplace equations with a critical growth term”, Differ. Integral Equ. 11 (1998),
no. 2, p. 311-326.
[2] Z. Chen, N. Shioji, W. Zou, “Ground state and multiple solutions for a critical exponent problem”, NoDEA, Nonlinear
Differ. Equ. Appl. 19 (2012), no. 3, p. 253-277.
[3] M. Degiovanni, S. Lancelotti, “Linking solutions for p-Laplace equations with nonlinearity at critical growth”, J.
Funct. Anal. 256 (2009), no. 11, p. 3643-3659.
[4] H. Egnell, “Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents”, Arch.
Ration. Mech. Anal. 104 (1988), no. 1, p. 57-77.
[5] J. P. Garcia-Azorero, I. Peral Alonso, “Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues”,
Commun. Partial Differ. Equations 12 (1987), no. 12, p. 1389-1430.
[6] M. Guedda, L. Véron, “Quasilinear elliptic equations involving critical Sobolev exponents”, Nonlinear Anal., Theory
Methods Appl. 13 (1989), no. 8, p. 879-902.
[7] S. Mosconi, K. Perera, M. Squassina, Y. Yang, “The Brezis–Nirenberg problem for the fractional p-Laplacian”, Calc.
Var. Partial Differ. Equ. 55 (2016), no. 4, article no. 105 (25 pages).
[8] K. Perera, “Nontrivial critical groups in p-Laplacian problems via the Yang index”, Topol. Methods Nonlinear Anal.
21 (2003), no. 2, p. 301-309.
[9] K. Perera, M. Squassina, Y. Yang, “Bifurcation and multiplicity results for critical fractional p-Laplacian problems”,
Math. Nachr. 289 (2016), no. 2-3, p. 332-342.
[10] ——— , “Bifurcation and multiplicity results for critical p-Laplacian problems”, Topol. Methods Nonlinear Anal. 47
(2016), no. 1, p. 187-194.
[11] A. Szulkin, T. Weth, M. Willem, “Ground state solutions for a semilinear problem with critical exponent”, Differ.
Integral Equ. 22 (2009), no. 9-10, p. 913-926.
C. R. Mathématique — 2021, 359, n 9, 1161-1164


On the existence of ground state solutions to critical growth problems nonresonant at zero

Othmane El Moize

Zouhaïr Mouayn

A $q$-deformation of true-polyanalytic Bargmann transforms when $q^{-1}>1$

Piotr Graczyk

Patrice Sawyer

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$

Stéphane R. Louboutin

On the continued fraction expansions of $(1+\protect \sqrt{pq})/2$ and $\protect \sqrt{pq}$

Hongjun Gao

Šárka Nečasová

Tong Tang

Comptes Rendus
Mathématique
Hongjun Gao, Šárka Nečasová and Tong Tang
On the hydrostatic approximation of compressible anisotropic
Navier–Stokes equations
Volume 359, issue 6 (2021), p. 639-644
<https://doi.org/10.5802/crmath.186>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 6, p. 639-644
https://doi.org/10.5802/crmath.186
Partial differential equations / Equations aux dérivées partielles
On the hydrostatic approximation of
compressible anisotropic Navier–Stokes
equations
Sur l’approximation hydrostatique des équations de
Navier–Stokes anisotropes compressibles
Hongjun Gaoa, Šárka Nečasováb and Tong Tangc
a School of Mathematics, Southeast University, Nanjing 211189, P.R. China
b Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
c School of Mathematical Science, Yangzhou University, Yangzhou 225002, P.R. China
E-mails: gaohj@hotmail.com, matus@math.cas.cz, tt0507010156@126.com
Abstract. In this work, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the
Navier–Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical con-
straint in general large scale motions meaning that the vertical scale is significantly smaller than horizontal.
We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier–
Stokes equations to the compressible Primitive Equations. This is the first work to use relative entropy in-
equality for proving hydrostatic approximation and derive the compressible Primitive Equations.
Résumé. Dans ce travail, nous obtenons l’approximation hydrostatique en prenant la limite du petit rapport
d’aspect des équations de Navier–Stokes. Le rapport d’aspect (le rapport de la profondeur à la largeur hori-
zontale) est une contrainte géométrique dans les mouvements géophysiques signifiant que l’échelle verticale
est nettement plus petite que l’horizontale. Nous utilisons l’inégalité d’entropie relative pour prouver rigou-
reusement la limite des équations de Navier–Stokes compressibles aux équations primitives compressibles.
C’est le premier travail qui utilise l’inégalité d’entropie relative pour prouver l’approximation hydrostatique
et dériver les équations primitives compressibles.
Funding. The research of H. G is partially supported by the NSFC Grant No. 11531006 and the Science Climb-
ing Program of Southeast University. The research of Š.N. is supported by the Czech Sciences Foundation
(GAČR), GA19-04243S and RVO 67985840. The research of T.T. is supported by the NSFC Grant No. 11801138.
Š.N. and T.T. were partially supported by the special Grant of Jiangsu Provincial policy guidance plan for in-
troducting foreign talents - BX2020082.
Manuscript received 17th October 2020, accepted 3rd February 2021.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
640 Hongjun Gao, Šárka Nečasová and Tong Tang
1. Introduction
In this paper, we consider the following compressible anisotropic Navier–Stokes equations{
∂tρ+div(ρu) = 0,
(ρu)t +div(ρu⊗u)+∇p(ρ) =µx∆x u+µz∂zz u,
(1)
in the thin domain (0,T )×Ωε. We considerΩε = {(x, z)|x ∈T2,−ε< z < ε}, x denotes the horizontal
direction and z denotes the vertical direction, while, µx and µz are given constant horizontal
viscous coefficient and vertical viscous coefficient. The velocity u = (v, w), where v(t , x, z) ∈ R2
and w(t , x, z) ∈R represent the horizonal velocity and vertical velocity respectively. The pressure
p(ρ) satisfies the barotropic pressure law where the pressure and the density are related by the
formula: p(ρ) = ργ (γ > 1). Through out this paper, we use divu = divx v+∂z w and ∇ = (∇x ,∂z )
to denote the three-dimensional spatial divergence and gradient respectively, and ∆x stands for
horizontal Laplacian. Similar to the assumptions by [1, 8], we suppose µx = 1 and µz = ε2. As was
stressed by Azérad and Guillén [1], it is necessary to consider the above anisotropic viscosities
scaling, which is the fundamental assumption for the derivation of Primitive Equations (PE).
Under this assumption, the system is rewritten as the following
∂tρ+divx (ρv)+∂z (ρw) = 0,
ρ∂t v+ρ(u ·∇)v−∆x v−ε2∂zz v+∇x p(ρ) = 0,
ρ∂t w +ρu ·∇w −∆x w−ε2∂zz w +∂z p(ρ) = 0.
(2)
Inspired by [1, 8] and following geophysical assumption such as Pedlosky [12], we introduce
the following new unknowns,
uε = (vε, wε), vε(x, z, t ) = v(x,εz, t ), wε = 1
ε
w(x,εz, t ), ρε = ρ(x, t ),
for any (x, z) ∈Ω := T2 × (−1,1). It was observed that atmosphere and ocean are the thin layers,
where the fluid layer depth is small compared to radius of sphere. Pedlosky [12] pointed out that
“the pressure difference between any two points of the same vertical line depends only on the
weight of the fluid between these points . . . ”. Here we neglect the gravity and suppose the pressure
satisfies the barotropic pressure law. Therefore we assume the density to be independent of the
vertical variable. Then the system (2) becomes the following compressible scaled Navier–Stokes
equations (CNS): 
∂tρε+divx (ρεvε)+∂z (ρεwε) = 0,
ρε∂t vε+ρε(uε ·∇)vε−∆x vε−∂zz vε+∇x p(ρε) = 0,
ε2(ρε∂t wε+ρεuε ·∇wε−∆x wε−∂zz wε)+∂z p(ρε) = 0.
(3)
We supplement the CNS with the following boundary and initial conditions:
ρε, uε are periodic in x, y, z,
(ρε,uε)|t=0 = (ρ0,u0). (4)
The goal of this work is to investigate the limit process ε→ 0 in the system of (3) converges in a
certain sense to the following compressible Primitive Equations(CPE):
∂tρ+divx (ρv)+∂z (ρw) = 0,
∂t (ρv)+divx (ρv⊗v)+∂z (ρvw)+∇x p(ρ) =∆x v+∂zz v,
∂z p(ρ) = 0.
(5)
The PE model is a fundamental model to understand the atmosphere and ocean. There
is a large literatures dedicated to PE model see [3, 9] and references therein. The hydrostatic
approximation is one of the important feature of PE model. Rigorous justification of the limit
C. R. Mathématique — 2021, 359, n 6, 639-644
Hongjun Gao, Šárka Nečasová and Tong Tang 641
passage from anisotropic Navier–Stokes equations to its hydrostatic approximation via the small
aspect limit seems to be of obvious practical importance. There are numerous studies of the
incompressible convergence. For example, Azérad and Guillén [1] proved the weak solutions
of anisotropic Navier–Stokes equation converge to weak solutions of PE. Then Li and Titi [8]
used the method of weak-strong uniqueness to prove the aspect ratio limit of incompressible
anisotropic Navier–Stokes equations, that is from weak solutions of anisotropic Navier–Stokes
equations to strong solutions of incompressible PE model. Then Giga, Hieber and Kashiwabara
et al. [6] extended the results into maximal regularity spaces. However, for the compressible
fluids flows, to the best of authors’ knowledge, there are no results concerning the convergence
from CNS to CPE. Our result generalizes the results by Azérad and Guillén [1] and Li and
Titi [8] in the incompressible case to compressible case. However the compressible case is very
different because one cannot derive information on the vertical velocity with the incompressible
constraint.
Our goal is to rigorously derive the limit in the framework of weak solutions of CNS. Recently,
Maltese and Novotný [11] proved the convergance from 3D compressible Navier–Stokes equa-
tions to 2D compressible Navier–Stokes equations in thin domain. See also result by Ducomet
et al. [4]. Heuristically, inspired by their works, we develop the corresponding idea of relative en-
tropy inequality for the compressible Navier–Stokes equations, and utilize the special structure of
CPE to prove the convergence. There are huge differences of mathematical structure between the
Navier–Stokes equations and the CPE model. Due to the hydrostatic approximation, there is no
information for the vertical velocity in the momentum equation of CPE model, and the vertical
velocity is determined by the horizontal velocity via the continuity equation, so it is very difficult
to analyze the CPE model. Therefore, the classical method used in Navier–Stokes system can not
be applied straightforwardly to CPE. This is the first work to use the relative entropy inequality
for proving an hydrostatic approximation at compressible case.
2. Main result
Before showing our main result, we give the definition of weak solutions for CNS and the strong
solution for CPE. Recently, Bresch and Jabin [2] consider different compactness method from
Lions or Feireisl which can be applied to the anisotropical stress tensor. They obtain the global
existence of weak solutions for non-monotone pressure. Let us recall their definitions here.
2.1. Dissipative weak solutions of CNS
Definition 1. We say that [ρε,uε] with uε = (vε, wε) is a finite energy weak solution to the system
of (3), supplemented with initial data (4) if ρε = ρε(x, t ) and
uε ∈ L2(0,T ; H 1(Ω)), ρ|uε|2 ∈ L∞(0,T ;L1(Ω)),
ρε ∈ L∞(0,T ;Lγ(Ω))∩C ([0,T ],L1(Ω)),
(6)
• the continuity equation[∫
Ω
ρεψdxdz
]t=τ
t=0
=
∫ τ
0
∫
Ω
ρε∂tψ+ρεvε ·∇xψ+ρεwε∂zψdxdzdt , (7)
holds for all ψ ∈C∞c ([0,T )×Ω);
C. R. Mathématique — 2021, 359, n 6, 639-644
642 Hongjun Gao, Šárka Nečasová and Tong Tang
• the momentum equation
[∫
Ω
ρεvεϕHdxdz
]t=τ
t=0
−
∫ τ
0
∫
Ω
ρεvε∂tϕHdxdzdt −
∫ τ
0
∫
Ω
ρεuεvε ·∇ϕHdxdzdt
+
∫ τ
0
∫
Ω
∇vε ·∇ϕHdxdzdt −
∫ τ
0
∫
Ω
p(ρε)divx ϕHdxdzdt = 0, (8)
and
ε2
[∫
Ω
ρεwεϕ3dxdz
]t=τ
t=0
−ε2
∫ τ
0
∫
Ω
ρεwε∂tϕ3dxdzdt −ε2
∫ τ
0
∫
Ω
ρεuεwε ·∇ϕ3dxdzdt
+ε2
∫ τ
0
∫
Ω
∇wε ·∇ϕ3dxdzdt −
∫ τ
0
∫
Ω
p(ρε)∂zϕ3dxdzdt = 0, (9)
holds for all ϕH,ϕ3 ∈C∞c ([0,T )×Ω). Combining (8)–(9), we obtain
[∫
Ω
ρεvεϕHdxdz +ε2
∫
Ω
ρεwεϕ3dxdz
]t=τ
t=0
−
∫ τ
0
∫
Ω
ρεvε∂tϕHdxdzdt
−ε2
∫ τ
0
∫
Ω
ρεwε∂tϕ3dxdzdt −
∫ τ
0
∫
Ω
ρεvε⊗vε : ∇xϕHdxdzdt
−
∫ τ
0
∫
Ω
ρεvεwε ·∂zϕH dxdzdt −ε2
∫ τ
0
∫
Ω
ρεvεwε ·∇xϕ3dxdzdt
−ε2
∫ τ
0
∫
Ω
ρεw
2
ε∂zϕ3dxdzdt +
∫ τ
0
∫
Ω
∇vε : ∇ϕHdxdzdt
+ε2
∫ τ
0
∫
Ω
∇wε ·∇ϕ3dxdzdt −
∫ τ
0
∫
Ω
p(ρε)divϕdxdzdt = 0, (10)
where ϕ= (ϕH,ϕ3) ∈C∞c ([0,T )×Ω) and divϕ= divH ϕx +∂zϕ3,
• the energy inequality
[∫
Ω
1
2
ρε|vε|2 + ε
2
2
ρε|wε|2 +P (ρε))dxdz
]t=τ
t=0
+
∫ τ
0
∫
Ω
(|∇vε|2 +ε2|∇wε|2)dxdzdt ≤ 0, (11)
holds for a.a τ ∈ (0,T ), where P (ρ) = ρ ∫ ρ1 p(z)z2 dz.
2.2. Strong solution of CPE
We say that (r,u), u = (v,W ) is a strong solution to the CPE system (5) in (0,T )×Ω, if
r
1
2 ∈ L∞(0,T ; H 2(Ω)), ∂t r
1
2 ∈ L∞(0,T ; H 1(Ω)), r > 0 for all (t , x),
v ∈ L∞(0,T ; H 3(Ω))∩L2(0,T ; H 4(Ω)), ∂t v ∈ L2(0,T ; H 2(Ω)),
with initial data r
1
2
0 ∈ H 2(Ω), r0 > 0 and v0 ∈ H 3(Ω). Liu and Titi [10] have proved the local
existence of strong solution to CPE system (5).
C. R. Mathématique — 2021, 359, n 6, 639-644
Hongjun Gao, Šárka Nečasová and Tong Tang 643
2.3. Relative entropy inequality
Motivated by [5], for any finite energy weak solution (ρ,u), where u = (v, w), to the CNS system,
we introduce the relative energy functional
E (ρ,u|r,u) =
∫
Ω
[
1
2
ρ|v−v|2 + ε
2
2
ρ|w −W |2 +P (ρ)−P ′(r )(ρ− r )−P (r )
]
dxdz
=
∫
Ω
(
1
2
ρ|v|2 + ε
2
2
ρ|w |2 +P (ρ)
)
dxdz −
∫
Ω
(ρv ·v+ε2ρwW )dxdz
+
∫
Ω
[
ρ
|v|2
2
+ ε
2
2
ρ|W |2 −P ′(r )
]
dxdz +
∫
Ω
p(r )dxdz
=
4∑
i=1
Ii , (12)
where r > 0, u = (v,W ) are smooth “test” functions, r , u compactly supported inΩ.
Lemma 2. Let (ρ,v, w) be a dissipative weak solution introduced in Definition 1. Then (ρ,v, w)
satisfy the relative entropy inequality
E (ρ,u|r,u)|t=τt=0 +
∫ τ
0
∫
Ω
(∇v · (∇v−∇v)+ε2|∇w |2)dxdzdt
≤
∫ τ
0
∫
Ω
ρ(∂t v+v∇x v+w∂z v)(v−v)dxdt
+ε2
∫ τ
0
∫
Ω
ρ(∂t W +v∇xW +w∂zW )(W −w)dxdzdt +ε2
∫ τ
0
∫
Ω
∇w ·∇W dxdzdt
−
∫ τ
0
∫
Ω
P ′′(r )((ρ− r )∂t r +ρv∇x r )dxdzdt −
∫ τ
0
∫
Ω
p(r )divx vdxdzdt . (13)
2.4. Main result
Now, we are ready to state our main result.
Theorem 3. Let γ> 4, Tmax > 0 be the life time of strong solution to CPE system (3) corresponding
to initial data [r0,v0]. Let (ρε,uε), uε = (vε, wε) be a sequence of dissipative weak solutions to the
CNS system (2) from the initial data (ρ0,ε,u0,ε). Suppose that
E (ρ0,ε,u0,ε|r0,u0) → 0,
where u0 = (v0,W0), then
ess sup
t∈(0,Tmax )
E (ρε,uε|r,u) → 0,
where u = (v,W ) and the couple (r,u) satisfy the CPE system (3) on the time interval [0,Tmax ).
For the proof see [7].
Remark 4. There is a constraint for the pressure index at Bresch and Jabin [2] that is γ >
3
2
( 4
3 +
p
10
3
)' 3.5. Thus, our condition satisfies their assumption.
Remark 5. It is important to point out that our convergence holds on a fixed time interval due
to the local existence of CPE. Some results [6, 8] concerning the incompressible PE model were
shown for the global convergence based on the global existence in time under assumptions on
the smallness of initial data.
C. R. Mathématique — 2021, 359, n 6, 639-644
644 Hongjun Gao, Šárka Nečasová and Tong Tang
Acknowledgements
The authors thank Prof. Edriss Titi’s wonderful and great talk on the course “compact course
Mathematical Analysis of Geophysical Models and Data Assimilation” held from 26 June to 10
July 2020, which offers insightful and constructive suggestions.
References
[1] P. Azérad, F. Guillén, “Mathematical justification of the hydrostatic approximation in the primitive equations of
geophysical fluid dynamics”, SIAM J. Math. Anal. 33 (2001), p. 847-859.
[2] D. Bresch, P.-E. Jabin, “Global existence of weak solutions for compressible Navier-Stokes equations: thermodynam-
ically unstable pressure and anisotropic viscous stress tensor”, Ann. Math. 188 (2018), no. 2, p. 577-684.
[3] C. Cao, E. S. Titi, “Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean
and atmosphere dynamics”, Ann. Math. 166 (2007), no. 1, p. 245-267.
[4] B. Ducomet, Š. Nečasová, M. Pokorný, M. A. Rodríguez-Bellido, “Derivation of the Navier-Stokes-Poisson system with
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[5] E. Feireisl, B. J. Jin, A. Novotný, “Relative entropies, suitable weak solutions, and weak-strong uniqueness for the
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[6] K. Furukawa, Y. Giga, M. Hieber, A. Hussein, T. Kashiwabara, M. Wrona, “Rigorous justification of the hydrostatic
approximation for the primitive equations by scaled Navier-Stokes equations”, https://arxiv.org/abs/1808.02410,
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[7] H. Gao, Š. Nečasová, T. Tang, “On the hydrostatic approximation of compressible anisotropic Navier–Stokes equa-
tions”, preprint Institute of Mathematics, 2020.
[8] J. Li, E. S. Titi, “The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: rigorous
justification of the hydrostatic approximation”, J. Math. Pures Appl. 124 (2019), p. 30-58.
[9] J. L. Lions, R. Temam, S. Wang, “New formulations of the primitive equations of atmosphere and applications”,
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[10] X. Liu, E. S. Titi, “Local well-posedness of strong solutions to the three-dimensional compressible Primitive Equa-
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[12] J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1979.
C. R. Mathématique — 2021, 359, n 6, 639-644


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