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ALGEBRAIC
 COMBINATORICS
Chi Hoi Yip
Maximality of subfields as cliques in Cayley graphs over finite fields
Volume 6, issue 4 (2023), p. 901-905.
https://doi.org/10.5802/alco.291
© The author(s), 2023.
This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Algebraic Combinatorics is published by The Combinatorics Consortium
and is a member of the Centre Mersenne for Open Scientific Publishing
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e-ISSN: 2589-5486
Algebraic Combinatorics
Volume 6, issue 4 (2023), p. 901–905
https://doi.org/10.5802/alco.291
Maximality of subfields as cliques in Cayley
graphs over finite fields
Chi Hoi Yip
Abstract We show the maximality of subfields as cliques in a special family of Cayley graphs
defined on the additive group of a finite field. In particular, this confirms a conjecture of Yip
on generalized Paley graphs.
1. Introduction
Throughout the paper, let p be an odd prime and q a power of p. Let Fq be the finite
field with q elements, F+q be its additive group, and F∗q = Fq∖{0} be its multiplicative
group.
In this paper we study maximal cliques in Cayley graphs. We begin by recalling
some basic terminologies. Given an abelian group G and a connection set S ⊂ G∖{0}
with S = −S, the Cayley graph Cay(G, S) is the undirected graph whose vertices are
elements of G, such that two vertices g and h are adjacent if and only if g − h ∈ S. A
clique in a graph X is a subset of vertices in X such that every two distinct vertices
in the clique are adjacent. The clique number of X, denoted by ω(X), is the size of a
maximum clique in X. A maximal clique is a clique that is not contained in a strictly
larger clique.
Generalized Paley graphs are well-studied Cayley graphs. They were first intro-
duced by Cohen [4] in 1988, and have been reintroduced by several groups of authors.
Let d > 1 be a positive integer and q ≡ 1 (mod 2d). The d-Paley graph on Fq, denoted
GP (q, d), is the Cayley graph Cay(F+q , (F∗q)d), where (F∗q)d is the set of d-th powers
in F∗q . Note that the condition q ≡ 1 (mod 2d) avoids degeneracy of the graph; see
for example [4, Section 4]. Note that Paley graphs are simply 2-Paley graphs. 3-Paley
graphs and 4-Paley graphs are also known as cubic Paley graphs and quadruple Paley
graphs.
It is known that in the Paley graph GP (q2, 2), the subfield Fq forms a maxi-
mal clique for a trivial reason [3]: the clique number of GP (q2, 2) is q. In general,
Broere, Döman, and Ridley [3] observed that in the generalized Paley graph GP (qn, d),
the subfield Fq forms a clique if d | q
n−1
q−1 . In this case, this observation leads to
ω(GP (qn, d)) ⩾ q, which is much better than the generic best-known lower bound
O(log q) on the clique number that holds for all generalized Paley graphs due to Co-
hen [4]. In fact, Green [6] showed that the clique number of almost all Cayley graphs
Manuscript received 6th September 2022, revised 8th January 2023, accepted 9th January 2023.
Keywords. Cayley graph, maximal clique, character sum.
Acknowledgements. The author was supported by a doctoral fellowship from the University of
British Columbia.
ISSN: 2589-5486 http://algebraic-combinatorics.org/
C. H. Yip
defined on a cyclic group G is O(log |G|), as |G| → ∞. While in the case of gen-
eralized Paley graphs, the underlying group may not be cyclic, Green’s result still
suggests that a clique in a Cayley graph with exceptional size (that is, much larger
than O(log q)) tends to have special algebraic structures.
Determining the clique number of (generalized) Paley graphs is widely open in
general [5]; we refer to [11, Section 1.3] for a survey on recent (minor) improvements
on the square root trivial upper bound. Thus, it is interesting if one can show that
the above subfield constructions of cliques are not maximal so that the lower bound
on the clique number can be further improved. However, the rigid algebraic structure
of subfields suggests that it is very unlikely. Indeed, in [12, Conjecture 1.4], Yip
conjectured that such constructions give rise to maximal cliques:
Conjecture 1.1 ([12, Conjecture 1.4]). Let d > 1 be an integer. Let q ≡ 1 (mod 2d)
be a power of a prime p, and let r be the largest integer such that d | q−1pr−1 . Then the
subfield Fpr forms a maximal clique in GP (q, d).
In other words, the conjecture states that if Fpr is the maximum subfield of Fq that
forms a clique in GP (q, d), then in fact it forms a maximal clique. The motivation of
Conjecture 1.1 is explained in [12] in greater details; in particular, Yip [12, Section 3]
showed that if Fpr is not maximal, then there is a clique that is a 2-dimensional space
over Fpr and consequently the lower bound on the clique number can be improved
significantly to ω(GP (q, d)) ⩾ p2r. This observation, together with known upper
bounds on the clique number, allows Yip [12, Theorem 1.5 and Theorem 1.6] to
confirm Conjecture 1.1 for cubic Paley graphs with cubic order and quadruple Paley
graphs with quartic order. However, a similar argument fails to work in general since
the best-known upper bound on the clique number is O(√q).
In this paper, we use different ideas to resolve Conjecture 1.1. For simplicity, we
call a clique C in a Cayley graph X = Cay(F+q , S) a subfield clique if C is a subfield
of Fq, and we say C is a maximal subfield clique if C is not contained in a strictly
larger subfield clique. Our first main result confirms Conjecture 1.1 in a stronger form:
a maximal subfield clique in a generalized Paley graph is a maximal clique.
Theorem 1.2. Let d > 1 be an integer. Let q be a prime power with qn ≡ 1 (mod 2d)
and q > (n − 1)2. If Fq is a maximal subfield clique in GP (qn, d), then Fq is also a
maximal clique.
In [12, Theorem 1.7], Yip described a similar phenomenon in Peisert graphs and
conjectured that Fq forms a maximal clique in a Peisert graph with order q4 provided
that q > 3; this was confirmed by Asgarli and Yip in [1, Theorem 1.5]. Moreover, in [1,
Section 5] of the same paper, they observed that a similar result holds for generalized
Peisert graphs under extra assumptions.
Our second main result improves and extends the results in [1, Section 5] substan-
tially. Before stating that, we shall recall the definition of generalized Peisert graphs,
first introduced by Mullin [9]. This definition is motivated by the similarity between
generalized Paley graphs and Peisert graphs (first introduced by Peisert in [10] in
order to classify self-complementary symmetric graphs).
Definition 1.3 ([1, Definition 2.11]). Let d be a positive even integer, and q a prime
power such that q ≡ 1 (mod 2d). The d-th power Peisert graph of order q, denoted
GP ∗(q, d), is the Cayley graph Cay(F+q , Mq,d), where
Mq,d =
{
gdk+j : 0 ⩽ j ⩽ d2 − 1, k ∈ Z
}
,
and g is a primitive root of Fq.
Algebraic Combinatorics, Vol. 6 #4 (2023) 902
Maximality of subfields as cliques in Cayley graphs
While the definition of GP ∗(q, d) depends on the choice of the primitive root g, it is
clear that the isomorphism class of GP ∗(q, d) is independent of the choice of g. We re-
fer to [1, Remark 2.12] for a discussion on the connection between generalized Peisert
graphs and generalized Paley graphs. In particular, if d is even, then GP ∗(q, d) con-
tains GP (q, d) as a subgraph and thus the structure of maximal cliques in GP ∗(q, d)
is potentially richer. However, our second main result shows that a maximal subfield
clique in GP ∗(q, d) is still a maximal clique.
Theorem 1.4. Let d ⩾ 4 be an even integer. Let q be a prime power such that
qn ≡ 1 (mod 2d) and q > (n − 1)2d4/π2(d − 1)2. If Fq is a maximal subfield clique
in GP ∗(qn, d), then Fq is also a maximal clique.
Note that Theorem 1.4 refines Theorem 1.2 provided that q is sufficiently large.
In fact, we will prove a more general (yet technical) statement for any Cayley graph
containing a generalized Paley graph as a subgraph in Proposition 3.1. Before proving
our main results, we shall introduce some preliminary tools in Section 2.
2. Preliminaries
A multiplicative character of Fq is a group homomorphism from F∗q to the multiplica-
tive group of complex numbers with modulus 1. For a multiplicative character χ, its
order d is the smallest positive integer such that χd = χ0, where χ0 is the trivial
multiplicative character of Fq. We refer to [8, Chapter 5] for a general discussion on
estimates on character sums. The following theorem, due to Katz [7], is crucial in our
proofs.
Theorem 2.1 (Katz). Let θ ∈ Fqn such that Fq(θ) = Fqn . Let χ be a non-trivial
multiplicative character of Fqn . Then∣∣∣∣∣∣
∑
a∈Fq
χ(θ + a)
∣∣∣∣∣∣ ⩽ (n − 1)√q.
The following definition is helpful for our discussions.
Definition 2.2 ([1, Definition 2.16]). Let ϵ > 0. A set M ⊂ C is said to be ϵ-lower
bounded if for every integer k ∈ N, and for every choice of x1, x2, . . . , xk ∈ M , we
have ∣∣∣∣ k∑
j=1
xj
∣∣∣∣ ⩾ ϵk.
Using trigonometric manipulations, it is not difficult to show the following lemma.
Lemma 2.3 ([1, Lemma 4.5]). Let d ⩾ 4 be an even integer, and ω = exp(2πi/d).
Then the set M = {ωj : 0 ⩽ j ⩽ d/2 − 1} is
(
π
d −
π
d2
)
-lower bounded.
3. Proof of main results
We will prove a more general statement in the following proposition, and then deduce
Theorem 1.2 and Theorem 1.4 as special cases.
Proposition 3.1. Let n ⩾ 2 be an integer and ϵ > 0 a real number. Let X =
Cay(F+qn , S) be a Cayley graph with q > (n − 1)2/ϵ2. Assume that there is an integer
d > 1, such that X contains GP (qn, d) as a subgraph and the set M = {χ(x) : x ∈ S}
is ϵ-lower bounded for some multiplicative character χ of Fqn with order d. If Fq is a
maximal subfield clique in X, then Fq is also a maximal clique in X.
Algebraic Combinatorics, Vol. 6 #4 (2023) 903
C. H. Yip
Proof. Assume that Fq is not a maximal clique; then we can find θ ∈ Fqn ∖ Fq such
that Fq ∪{θ} forms a clique in X. Thus, by definition, for any a ∈ Fq, we have θ−a ∈ S
and χ(θ − a) ∈ M .
Let Fqm be the smallest extension of Fq that contains θ; then Fqm is necessarily a
subfield of Fqn and m > 1. Let χ′ be the restriction of χ on the subfield Fqm ; then χ′
is a multiplicative character of Fqm .
Suppose χ′ is the trivial multiplicative character of Fqm ; then χ(x) = 1 for each
x ∈ F∗qm . This means that each element in F∗qm is a d-th power in F∗qn and it follows
that F∗qm ⊂ S since X contains GP (qn, d) as a subgraph. In particular, Fqm is a
subfield clique in X that is strictly larger than Fq, violating the assumption. Thus,
χ′ is a non-trivial multiplicative character of Fqm .
Applying Theorem 2.1 to the character χ′ on the affine line θ + Fq and using the
definition that M is ϵ-lower bounded, we obtain that
ϵq ⩽
∣∣∣∣∣∣
∑
a∈Fq
χ(θ + a)
∣∣∣∣∣∣ =
∣∣∣∣∣∣
∑
a∈Fq
χ′(θ + a)
∣∣∣∣∣∣ ⩽ (m − 1)√q ⩽ (n − 1)√q.
Therefore, q ⩽ (n − 1)2/ϵ2, contradicting our assumption. This shows that Fq is a
maximal clique in X. □
Remark 3.2. From the proof, it is easy to see that the condition “{χ(x) : x ∈ S}
is ϵ-lower bounded" in the statement of the above proposition can be weakened to
|
∑
x∈A χ(x)| ⩾ ϵq for any A ⊂ S with |A| = q. In other words, if there is S′ ⊂ S
such that |S ∖ S′| is small and {χ(x) : x ∈ S′} is ϵ-lower bounded, then we can still
conclude that Fq is maximal clique provided that q is sufficiently large.
Remark 3.3. A result of a similar flavor has appeared in [1, Theorem 1.3] in terms
of maximum cliques in the so-called Peisert-type graphs. It generalizes the celebrated
Van Lint–MacWilliams’ conjecture (equivalently, Erdős–Ko–Rado theorem for Paley
graphs of square order), first proved by Blokhuis [2]. We refer to [1, Section 2] for a
historical discussion.
Finally, we prove Theorem 1.2 and Theorem 1.4, and discuss the sharpness of the
assumption that q is sufficiently large in both theorems.
Proof of Theorem 1.2. Note that the connection set S of GP (qn, d) consists of d-th
powers in F∗qn . It follows that M = {χ(x) : x ∈ S} = {1} is 1-lower bounded for any
multiplicative character χ of Fqn with order d. Thus, the theorem follows immediately
from Proposition 3.1. □
Remark 3.4. We conjecture that the condition q > (n − 1)2 in Theorem 1.2 can
be dropped. However, we do not know how to remove this condition. When n ⩽ 5,
we verified that Theorem 1.2 holds for all q ⩽ (n − 1)2 by enumerating all possible
generalized Paley graphs via SageMath. We also verified that Theorem 1.2 holds for
all q ⩽ 17 when n = 6.
Proof of Theorem 1.4. Let g be the primitive root of Fqn that defines the graph
GP ∗(qn, d). Let χ be a multiplicative character in Fq such that χ(g) = ω, where ω =
exp(2πi/d); then χ has order d. As discussed before, GP ∗(qn, d) contains GP (qn, d)
as a subgraph. Let M = {χ(x) : x ∈ S}, where S = {gj+kd : 0 ⩽ j ⩽ d/2 − 1, k ∈ Z}
is the connection set of GP ∗(qn, d). It then follows from Lemma 2.3 that the set
Algebraic Combinatorics, Vol. 6 #4 (2023) 904
Maximality of subfields as cliques in Cayley graphs
M = {ωj : 0 ⩽ j ⩽ d/2 − 1} is
(
π
d −
π
d2
)
-lower bounded. Therefore, by Proposi-
tion 3.1, Fq is a maximal clique provided that
q >
(n − 1)2(
π
d −
π
d2
)2 = (n − 1)2d4π2(d − 1)2 .
□
Remark 3.5. We believe that the condition q > (n−1)2d4/π2(d−1)2 in Theorem 1.4
is not optimal. However, we do need to assume q is sufficiently large for Theorem 1.4
to hold. There are plenty of counterexamples when q is small compared to n and d.
For example, when q = 3, n = 4, and d = 4, the subfield F3 is a maximal subfield
clique in GP ∗(81, 4), and yet there is a maximal clique with size 9 containing F3.
Similarly, when q = 5, n = 6, and d = 62, the subfield F5 is a maximal subfield clique
in GP ∗(15625, 62), and yet there is a maximal clique with size 25 containing F5.
Acknowledgements. The author thanks Shamil Asgarli for many helpful discussions.
References
[1] Shamil Asgarli and Chi Hoi Yip, Van Lint–MacWilliams’ conjecture and maximum cliques
in Cayley graphs over finite fields, J. Combin. Theory Ser. A 192 (2022), article no. 105667
(23 pages).
[2] A. Blokhuis, On subsets of GF(q2) with square differences, Nederl. Akad. Wetensch. Indag.
Math. 46 (1984), no. 4, 369–372.
[3] I. Broere, D. Döman, and J. N. Ridley, The clique numbers and chromatic numbers of certain
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[4] Stephen D. Cohen, Clique numbers of Paley graphs, Quaestiones Math. 11 (1988), no. 2, 225–
231.
[5] Ernest S. Croot, III and Vsevolod F. Lev, Open problems in additive combinatorics, in Additive
combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007,
pp. 207–233.
[6] Ben Green, Counting sets with small sumset, and the clique number of random Cayley graphs,
Combinatorica 25 (2005), no. 3, 307–326.
[7] Nicholas M. Katz, An estimate for character sums, J. Amer. Math. Soc. 2 (1989), no. 2, 197–200.
[8] Rudolf Lidl and Harald Niederreiter, Finite fields, second ed., Encyclopedia of Mathematics and
its Applications, vol. 20, Cambridge University Press, Cambridge, 1997, With a foreword by P.
M. Cohn.
[9] Natalie Mullin, Self-complementary arc-transitive graphs and their imposters, Master’s thesis,
University of Waterloo, 2009, https://uwspace.uwaterloo.ca/handle/10012/4264.
[10] Wojciech Peisert, All self-complementary symmetric graphs, J. Algebra 240 (2001), no. 1, 209–
229.
[11] Chi Hoi Yip, Gauss sums and the maximum cliques in generalized Paley graphs of square order,
Funct. Approx. Comment. Math. 66 (2022), no. 1, 119–138.
[12] , On maximal cliques of Cayley graphs over fields, J. Algebraic Combin. 56 (2022), no. 2,
323–333.
Chi Hoi Yip, Department of Mathematics, University of British Columbia, 1984 Mathematics Road,
Vancouver V6T 1Z2, Canada
E-mail : kyleyip@math.ubc.ca
Algebraic Combinatorics, Vol. 6 #4 (2023) 905


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Comptes Rendus
Mathématique
Dandan Ma and Fan Wu
Shinbrot’s energy conservation criterion for the 3D
Navier–Stokes–Maxwell system
Volume 361 (2023), p. 91-96
https://doi.org/10.5802/crmath.379
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
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 91-96
https://doi.org/10.5802/crmath.379
Partial differential equations / Equations aux dérivées partielles
Shinbrot’s energy conservation criterion for
the 3D Navier–Stokes–Maxwell system
DandanMaa and FanWu∗, a
a College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330099,
China
E-mails: 981423315@qq.com, wufan0319@yeah.net
Abstract. This paper concerns the energy conservation for the weak solutions to the Navier–Stokes–Maxwell
system. Although the Maxwell equation with hyperbolic nature, we still establish a Lq Lp type condition guar-
antee validity of the energy equality for the weak solutions. We mention that there no regularity assumption
on the electric field E .
2020 Mathematics Subject Classification. 76W05, 35Q30, 35Q61.
Manuscript received 24 January 2022, accepted 9 June 2022.
1. Introduction
In this paper, we study the incompressible Navier–Stoke–Maxwell system
∂t u +u ·∇u −µ∆u =−∇p + j ×B , divu = 0,
1
c ∂t E −∇×B =− j , j =σ(cE +u ×B),
1
c ∂t B +∇×E = 0, divB = 0,
(1)
with the initial data
u(0, x) = u0,E(0, x) = E0,B(0, x) = B0 (2)
for (t , x) ∈ R+ ×Ω, where Ω = T3 is a periodic domain in R3. Here c > 0 denotes the speed of
light, µ > 0 is the viscosity of the fluid, and σ > 0 is the electrical conductivity. In the above
system (1), u = (u1,u2,u3) = u(t , x) stands for the velocity field of the (incompressible) fluid,
while E = (E1,E2,E3) = E(t , x) and B = (B1,B2,B3) = B(t , x) are the electric and magnetic fields,
respectively. The scalar function p = p(t , x) is the pressure and is also an unknown. Observe,
though, that the electric current j = j (t , x) is not an unknown, for it is fully determined by (u,E ,B)
through Ohm’s law. For simplicity, we will take µ= c =σ= 1.
The Navier–Stokes–Maxwell system (1) describes the evolution of a plasma (i.e., a charged
fluid) subject to a self-induced electromagnetic Lorentz force j × B . Mathematically, (1) is a
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
92 Dandan Ma and Fan Wu
coupled system, constituted by the parabolic nature of the Navier–Stokes equations from fluid
dynamics and the hyperbolic of the Maxwell equation from electromagnetism. Moreover, it can
be derived from the Vlasov–Maxwell–Boltzmann system [2]. In the 2D case, Masmoudi [8] proved
global existence of regular solutions to the Maxwell–Navier–Stokes system (1), and also provide an
exponential growth estimate for the H s norm of the solution when the time goes to infinity. In the
3D case, Ibrahim–Keraani [10] showed the existence of global small mild solutions of system (1).
Recently, Arsenio–Gallagher [1] have made important and new progress in this direction, they
established that global existence of solutions to the 3D system (1) holds whenever the initial
datum (u0,E0,B0) is chosen in the natural energy space L2, while the electromagnetic field
(E0,B0) alone lies in Ḣ s , for some given s ∈ [ 12 , 32 ), and is sufficiently small when compared to
some non-linear function of the initial energy
E0 := 1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) .
Before discussing the contents of this paper let us recall some well-established facts regarding
the incompressible Navier–Stokes equations (corresponding to the case when (E ,B) = 0 in (1)), in
relation with this work. Pioneering works of Leray [5] and Hopf [3] showed the global existence of
a weak solution called Leray–Hopf solution with initial data u0 ∈ L2(Ω). As usual, a weak solution
u satisfies the energy inequality
‖u(t )‖2L2 +2
∫ T
0
∫
Ω
|∇u|2 d xd t ≤ ‖u0‖2L2 (3)
for any t ∈ (0,T ). On the other hand, regular solutions to the 3D Navier–Stokes equations satisfy
the energy equality:
‖u(t )‖2L2 +2
∫ T
0
∫
Ω
|∇u|2 d xd t = ‖u0‖2L2 . (4)
A natural question that still remains open is whether energy equality, which should be expected
from a physical point of view, is valid for weak solutions. Lions [6] and Ladyzhenskaya [4]
proved independently that such solutions satisfy the (global) energy equality (4) under the
additional assumption u ∈ L4L4. Shinbrot [9] generalized the Lions–Ladyzhenskaya condition to
u ∈ Lr (0,T ;Ls (Ω)) with 2r + 2s ≤ 1, s ≥ 4. Recently, Tan and Yin [12] proved that u ∈ L2,∞(0,T ;B MO)
implies energy equality (4).
For system (1), by results of Arsenio–Gallagher [1], it is know that for initial data (u0,E0,B0),
with divu0 = divB0 = 0, belongs to(
L2 × (H s)2)(R3) , 1
2
≤ s < 3
2
,
and
‖(E0,B0)‖Ḣ s C∗E
s− 12
0 e
C∗E0 ≤ 1,
then there is a global weak solution (u,E ,B) to the 3D Navier–Stokes–Maxwell system (1) such
that
u ∈ L∞ (R+;L2)∩L2 (R+; Ḣ 1) , (E ,B) ∈ L∞ (R+;L2) , j ∈ L2 (R+;L2) (5)
satisfying the energy inequality, for almost all t > 0,
1
2
(‖u(t )‖2L2 +‖E(t )‖2L2 +‖B(t )‖2L2)+∫ t
0
(‖∇u(τ)‖2L2 +‖ j (τ)‖2L2)dτ
≤1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) . (6)
Dandan Ma and Fan Wu 93
Moreover, the regular solutions to (1) satisfy the corresponding energy equality
1
2
(‖u(t )‖2L2 +‖E(t )‖2L2 +‖B(t )‖2L2)+∫ t
0
(‖∇u(τ)‖2L2 +‖ j (τ)‖2L2)dτ
=1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) . (7)
Although we can not prove that the weak solution is regular on time, due to the fact that the
Navier–Stoke equations is a sub-system of (1). An interesting question is whether or not the
energy equality (7) is holds for such weak solutions. By using the lemma introduced by Lions,
which has been used in the Yu’s article [13], we shall give an positive answer for this problem if
(u,B) ∈ Lq Lp .
We state our main result as follows:
Theorem 1. Let u ∈ L∞(0,T ;L2(Ω)) ∩ L2(0,T ; H 1(Ω)), (E ,B) ∈ L∞(0,T ;L2(Ω)) and j ∈ L2(0,T ;
L2(Ω)) be a weak solution to system (1). In addition, if u ∈ Lq (0,T ;Lp (Ω)) for any 1q + 1p
≤ 12 , p ≥ 4, and B ∈ Lr (0,T ;Ls (Ω)) for any 1r + 1s ≤ 12 , s ≥ 4, then∫
Ω
|u(t , x)|2 +|E(t , x)|2 +|B(t , x)|2 d x +2
∫ T
0
∫
Ω
|∇u|2 +| j |2d xd t
=
∫
Ω
|u0|2 +|E0|2 +|B0|2 d x
for any t ∈ [0,T ].
Remark 2. This result extends the well-known Shinbrot’s energy conservation criterion to the
Navier–Stokes–Maxwell system (1).
Remark 3. By means of arguement in [11], it is interesting to generalize Theorem 1 from periodic
domain to bounded Lipschitz domain.
2. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1. By using interpolation inequalities∥∥ f ∥∥L4(0,T ;L4(Ω)) ≤C ∥∥ f ∥∥θL∞(0,T ;L2(Ω)) ∥∥ f ∥∥1−θLn (0,T ;Lm (Ω)) ≤C
with 1n + 1m = 12 ,m > 4, we need only prove the case p = q = r = s = 4. For the sake of simplicity,
we will proceed as if the solution is differentiable in time. The extra arguments needed to mollify
in time are straightforward.
Let η : R3 → R be a standard mollifier, i.e. η(x) = C e
1
|x|2−1 for |x| < 1 and η(x) = 0 for |x| > 1,
where constant C > 0 selected such that ∫R3 η(x)dx = 1. For any ε > 0, we define the rescaled
mollifier ηε(x) = ε−3η
( x
ε
)
. For any function f ∈ L1loc(Ω), its mollified version is defined as
f ε(x) = ( f ∗ηε) (x) = ∫
Ω
ηε(x − y) f (y)dy.
If f ∈W 1,p (Ω), the following local approximation is well known
f ε(x) → f in W 1,pl oc (Ω) ∀ p ∈ [1,∞).
The crucial ingredient to prove Theorem 1 is the following important lemma proved by Lions [7]
Lemma 4. Let ∂ be a partial derivative in one direction. Let f ,∂ f ∈ Lp (R+ ×Ω), g ∈ Lq (R+ ×Ω)
with 1 ≤ p, q ≤∞, and 1p + 1q ≤ 1. Then, we have∥∥∂( f g )∗ηε−∂( f (g ∗ηε))∥∥Lr (R+×Ω) ≤C ∥∥∂ f ∥∥Lp (R+×Ω) ‖g‖Lq (R+×Ω)
for some constant C > 0 independent of ε, f and g , and with 1r = 1p + 1q . In addition,
94 Dandan Ma and Fan Wu
∂( f g )∗ηε−∂
(
f
(
g ∗ηε
))→ 0 in Lr (R+×Ω)
as ε→ 0, if r <∞.
With Lemma 4 in hand, we are ready to prove our main result. First, applying the mollifier to
the system (1) and multiplying all equations of the system by uε, Eε and Bε, respectively, one has
∫
Ωu
ε
(
∂t u +u ·∇u −∆u +∇p − j ×B
)εd x = 0,∫
ΩE
ε
(
∂t E −∇×B + j
)εd x = 0,∫
ΩB
ε (∂t B +∇×E)εd x = 0.
(8)
This yields
1
2
d
d t
∫
Ω
(∣∣uε∣∣2 + ∣∣Eε∣∣2 + ∣∣Bε∣∣2)d x +∫
Ω
∣∣∇uε∣∣2 d x
=−
∫
Ω
div(u ⊗u)ε ·uεd x +
∫
Ω
( j ×B)ε ·uεd x +
∫
Ω
(∇×B)ε ·Eεd x
−
∫
Ω
j ε ·Eεd x −
∫
Ω
(∇×E)ε ·Bεd x.
(9)
Clearly,∫
Ω
(∣∣uε∣∣2 + ∣∣Eε∣∣2 + ∣∣Bε∣∣2)d x −∫
Ω
(∣∣uε0∣∣2 + ∣∣Eε0∣∣2 + ∣∣Bε0 ∣∣2)d xd t +2∫ T
0
∫
Ω
∣∣∇uε∣∣2 d xd t
=−2
∫ T
0
∫
Ω
div(u ⊗u)ε ·uεd xd t +2
∫ T
0
∫
Ω
( j ×B)ε ·uεd xd t +2
∫ T
0
∫
Ω
(∇×B)ε ·Eεd xd t
−2
∫ T
0
∫
Ω
j ε ·Eεd xd t −2
∫ T
0
∫
Ω
(∇×E)ε ·Bεd xd t
=I1 + I2 + I3 + I4 + I5.
(10)
Notice that
div(u ⊗u)ε = [div(u ⊗u)ε−div(u ⊗uε)]+ [div(u ⊗uε)−div(uε⊗uε)]+div(uε⊗uε)
= I11 + I12 + I13,
thus
I1 =−2
∫ T
0
∫
Ω
(I11 + I12 + I13) ·uεd xd t . (11)
From Lemma 4, one obtains
‖I11‖
L
4
3
(
0,T ;L
4
3 (Ω)
) ≤C‖u‖L4(0,T ;L4(Ω))‖∇u‖L2(0,T ;L2(Ω))
and it converges to zero in L
4
3 (0,T ;L
4
3 (Ω)) as ε tends to zero. Thus, as ε goes to zero, it follows that∣∣∣∣−2∫ T
0
∫
Ω
I11u
εd xd t
∣∣∣∣= ∣∣∣∣∫ T
0
∫
Ω
[
div(u ⊗u)ε−div(u ⊗uε)] ·uεd xd t ∣∣∣∣
≤ ‖I11‖
L
4
3
(
0,T ;L
4
3 (Ω)
) ∥∥uε∥∥L4(0,T ;L4(Ω))
→ 0.
(12)
Dandan Ma and Fan Wu 95
Moreover, we have∣∣∣∣−2∫ T
0
∫
Ω
I12u
εd xd t
∣∣∣∣= ∣∣∣∣∫ T
0
∫
Ω
[
div
(
u ⊗uε)−div(uε⊗uε)] ·uεd xd t ∣∣∣∣
=
∣∣∣∣∫ T
0
∫
Ω
(
u ⊗uε−uε⊗uε) ·∇uεd xd t ∣∣∣∣
≤ ∥∥u −uε∥∥L4(0,T ;L4(Ω)) ∥∥uε∥∥L4(0,T ;L4(Ω)) ∥∥∇uε∥∥L2(0,T ;L2(Ω))
→ 0.
(13)
Since divuε = 0, one has
−2
∫ T
0
∫
Ω
I13 ·uεd x = 0. (14)
Combining (11)-(14), we know that
I1 → 0, as ε→ 0. (15)
For the term I2, we claim that
I2 = 2
∫ T
0
∫
Ω
( j ×B)ε ·uεd xd t → 2
∫ T
0
∫
Ω
( j ×B) ·ud xd t , as ε→ 0. (16)
Indeed,
2
∫ T
0
∫
Ω
( j ×B)ε ·uε− ( j ×B) ·ud xd t
=2
∫ T
0
∫
Ω
( j ×B)ε ·uε− ( j ×B) ·uε+ ( j ×B) ·uε− ( j ×B) ·ud xd t
=2
∫ T
0
∫
Ω
[
( j ×B)ε− ( j ×B)] ·uε+ ( j ×B) · (uε−u)d xd t
≤2∥∥( j ×B)ε− ( j ×B)∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ‖uε‖L4(0,T ;L4(Ω))
+2∥∥ j ×B∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ‖uε−u‖L4(0,T ;L4(Ω))
→ 0, as ε→ 0,
(17)
where used the fact that∥∥ j ×B∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ≤ ∥∥ j∥∥L2(0,T ;L2(Ω)) ‖B‖L4(0,T ;L4(Ω)) ≤C .
By using the same trick, we find that
I4 =−2
∫ T
0
∫
Ω
j ε ·Eεd xd t →−2
∫ T
0
∫
Ω
j ·E d xd t , as ε→ 0. (18)
After integration by parts, the term I3 can be dominated as
I3 =2
∫ T
0
∫
Ω
(∇×B)ε ·Eεd xd t
=2
∫ T
0
∫
Ω
(
εi j k∂ j Bk
)ε ·Eεi d xd t
=−2
∫ T
0
∫
Ω
εi j k B
ε
k ·∂ j Eεi d xd t
=2
∫ T
0
∫
Ω
εk j i B
ε
k ·∂ j Eεi d xd t
=2
∫ T
0
∫
Ω
Bε · (∇×E)εd xd t ,
(19)
96 Dandan Ma and Fan Wu
So, it follows that
I3 + I5 = 0. (20)
Letting ε go to zero in (10), and using (15)-(20), what we have proved is that in the limit∫
Ω
(|u|2 +|E |2 +|B |2)d x +2∫ T
0
∫
Ω
|∇u|2d xd t
=
∫
Ω
(|u0|2 +|E0|2 +|B0|2)d xd t +2∫ T
0
∫
Ω
( j ×B) ·ud xd t −2
∫ T
0
∫
Ω
j ·E d xd t . (21)
Thus, it holds that∫
Ω
(|u|2 +|E |2 +|B |2)d x +2∫ T
0
∫
Ω
|∇u|2 +| j |2d xd t =
∫
Ω
(|u0|2 +|E0|2 +|B0|2)d xd t , (22)
where we have used the fact that
2
∫ T
0
∫
Ω
( j ×B) ·ud xd t −2
∫ T
0
∫
Ω
j ·E d xd t =−2
∫
Ω
(u ×B) · j dx −2
∫
Ω
E · j dx
and
j = E +u ×B.
The proof of Theorem 1 is completed.
References
[1] D. Arsénio, G. Isabelle, “Solutions of Navier-Stokes-Maxwell systems in large energy spaces”, Trans. Am. Math. Soc.
373 (2020), no. 6, p. 3853-3884.
[2] R. Duan, “Green’s function and large time behavior of the Navier-Stokes-Maxwell system”, Anal. Appl., Singap. 10
(2012), no. 2, p. 133-197.
[3] E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen”, Math. Nachr. 4 (1951), no. 1-
6, p. 213-231.
[4] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and quasi-linear equations of parabolic type, Transla-
tions of Mathematical Monographs, vol. 23, American Mathematical Society, 1968.
[5] J. Leray, “Sur le mouvement d’un liquide visqueux emplissant l’espace”, Acta Math. 63 (1934), no. 1, p. 193-248.
[6] J.-L. Lions, “Sur la régularité et l’unicité des solutions turbulentes des équations de Navier-Stokes”, Rend. Semin.
Mat., Torino 30 (1960), p. 16-23.
[7] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models., Oxford Lecture Series in Mathe-
matics and its Applications, vol. 3, Oxford University Press, 1996.
[8] N. Masmoudi, “Global well posedness for the Maxwell-Navier-Stokes system in 2D”, J. Math. Pures Appl. 93 (2010),
no. 6, p. 559-571.
[9] M. Shinbrot, “The energy equation for the Navier-Stokes system”, SIAM J. Math. Anal. 5 (1975), no. 6, p. 948-954.
[10] I. Slim, K. Sahbi, “Global small solutions for the Navier-Stokes-Maxwell system”, SIAM J. Math. Anal. 43 (2011), no. 5,
p. 2275-2295.
[11] W. Tan, F. Wu, “On the energy equality for the 3D incompressible viscoelastic flows”, https://arxiv.org/abs/2111.
13547v1, 2021.
[12] W. Tan, Z. Yin, “The energy conservation and regularity for the Navier-Stokes equations”, https://arxiv.org/abs/2107.
04157, 2021.
[13] C. Yu, “A new proof to the energy conservation for the Navier-Stokes equations”, https://arxiv.org/abs/1604.05697,
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