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A new regularity criterion for weak solutions to the 3D micropolar fluid flows in terms of the pressure
Sadek Gala, Maria Alessandra Ragusa, Michel Thera
To cite this version:
Sadek Gala, Maria Alessandra Ragusa, Michel Thera. A new regularity criterion for weak solutions
to the 3D micropolar fluid flows in terms of the pressure. 2020. �hal-02651131�
(will be inserted by the editor)
A new regularity criterion for weak solutions to the 3D micropolar fluid flows in terms of the pressure
Sadek Gala · Maria Alessandra Ragusa · Michel Th´era
Received: date / Accepted: date
Abstract This note aims to giving a new regularity criterion for weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the field of pressure.
Keywords Micropolar fluid flows;·weak solutions;·Pressure criterion·Besov spaces..
Mathematics Subject Classification (2000) 35Q35·35B65·76D05
In this note we consider the following Cauchy problem (1) for the incompressible micropolar fluid equations inR3:
∂tu−∆u+ (u·∇)u+∇π−∇×ω=0, (1a)
∂tω−∆ ω−∇∇·ω+2ω+ (u·∇)ω−∇×u=0, (1b)
∇·u=0, (1c) u(x,0) =u0(x),ω(x,0) =ω0(x), (1d) whereu=u(x,t)∈R3,ω=ω(x,t)∈R3andπ=π(x,t)denote the unknown velocity of the fluid, the micro-rotational velocity of the fluid particles and the unknown scalar pressure of the fluid at the point(x,t)∈R3×(0,T), respectively, while u0,ω0are given initial data satisfying∇·u=0 in the sense of distributions.
This model for micropolar fluid flows proposed by Eringen [6] enables to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equa- tions for the viscous incompressible fluids, such as for example, the motion of animal blood, muddy fluids, liquid crystals and dilute aqueous polymer solutions, colloidal suspensions, etc.
When the micro-rotation effects are neglected or ω =0, (1) reduces to the in- compressible Navier-Stokes equations, and it is well known that regularity criteria
Sadek Gala ORCID 0000-0002-4286-4689
University of Mostaganem, P. O. Box 270, Mostaganem 27000, Algeria and Dipartimento di Matematica e Informatica, Universit`a di Catania
E-mail: sgala793@gmail.com
Maria Alessandra Ragusa ORCID 0000-0001-6611-6370
Dipartimento di Matematica e Informatica, Universit`a di Catania and RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia
E-mail: maragusa@dmi.unict.it
Michel Th´era ORCID 0000-0001-9022-6406
Mathematics and Computer Science Department, University of Limoges and Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology, Federation University Australia, Ballarat
E-mail: michel.thera@unilim.fr, m.thera@federation.edu.au
2 Sadek Gala et al.
for weak solution to the fluid dynamical models attracts more and more attention.
Velocity or vorticity or pressure blow-up criteria for Navier-Stokes equations, mi- cropolar fluid equations and magnetohydrodynamics (MHD) equations and so on (see e.g., [1–4,7–10,15–17] and the references therein) attracted during the last years the attention of many researchers.
As for the pressure criterion, let us first recall some results on pressure regularity of Navier-Stokes equations. In [14], He and Gala proved regularity of weak solutions under the condition
Z T 0
kπ(·,t)k2˙
B−1∞,∞dt<∞. (2)
Here and thereafter, ˙B−1∞,∞stands for the homogeneous Besov space, (for the definition see e.g. [14] and [13]). Later on, Guo and Gala [13] refined the condition (2) to
Z T 0
kπ(·,t)k2˙
B−1∞,∞
1+log
e+kπ(·,t)kB˙−1
∞,∞
dt<∞. (3) Motivated by the paper of Guo and Gala [13], the aim of this paper is to give a new regularity criterion for weak solutions to the 3D micropolar fluid flows in terms of the pressure in critical Besov spaces.
1 Main result
Let us start by stating the main result of this note.
Theorem 1.1 Let T >0and(u0,ω0)∈L2(R3)∩L4(R3)with∇·u0=0in the sense of distributions. Assume that(u,ω)is a weak solution to the 3D micropolar fluid flows (1) on(0,T). If the pressureπsatisfies the following condition :
Z T 0
kπ(·,t)k2˙
B−1∞,∞
e+log
e+kπ(·,t)kB˙−1
∞,∞
log
e+log
e+kπ(·,t)kB˙−1
∞,∞
dt<∞, (4) then(u,ω)is regular on(0,T], i.e.,(u,ω)∈C∞(R3×(0,T]).
Remark 1.1 This result provides a new information concerning the question of the regularity of weak solutions to the micropolar fluid equations and extends those of [14]
and [13]. In particular, the double-logarithm estimate (4) is sharper than any other results [13,14].
Before stating our result, let us recall what we mean by a weak solution.
Definition 1.1 ( [16])Let(u0,ω0)∈L2 R3
and suppose that∇·u0=0. A measur- able function (u(x,t),ω(x,t))is called a weak solution to the 3Dmicropolar flows equations (1) on(0,T)if(u,ω)satisfies three properties :
(1) (u,ω)∈L∞ (0,T);L2 R3
∩L2 (0,T);H1 R3
for allT>0;
(2) (u(x,t),ω(x,t))verifies (1) in the sense of distribution;
(3) For all 0≤t≤T it holds : ku(·,t)k2L2+kω(·,t)k2L2+2
Z t 0
(k∇u(·,τ)k2L2+k∇ω(·,τ)k2L2+k∇·ω(·,τ)k2L2)dτ
≤ ku0k2L2+kω0k2L2,
By a strong solution we mean a weak solution(u,ω)such that (u,ω)∈L∞ (0,T);H1 R3
∩L2 (0,T);H2 R3 .
It is well known that strong solutions are regular (say, classical) and unique in the class of weak solutions.
In order to prove Theorem1.1, we first establish some estimates between pres- sure and velocity. Taking div and∇div to both sides of the micropolar fluid flows for smooth solution(u,π), separately, we get the well-known pressure-velocity relation inR3, given by
π= (−∆)−1
3 i,
∑
j=1∂2
∂xi∂xj(uiuj) and ∇π= (−∆)−1
3 i,
∑
j=1∂2
∂xi∂xj(∇(uiuj)).
Then, the Calder´on-Zygmund inequality implies that for any 1<α<+∞:
kπkLα ≤Ckuk2L2α and k∇πkLα≤Ck |u|∇ukLα. (5)
2 Proof of Theorem1.1
Now we are in the position to prove Theorem1.1. Firstly, by means of the local exis- tence result, which is similar to the one used in the theory of Navier-Stokes equations (refer to Giga [12], see also Dong et al. [5]), and the standard local solution extension technique, equation (1) with(u0,ω0)∈L2(R3)∩L4(R3)admits a unique L4-strong solution(u,ω)on a maximal time interval. For the simplicity notation, we may sup- pose that the maximal time interval is[0,T). Thus, in order to prove Theorem1.1, it remains to show that
t→Tlim(ku(t)k4+kw(t)k4)<∞.
This will lead to a contradiction to the estimates to be derived below.
Proof Before going into the proof, we recall the following inequality established in [11] (see also [13]):
kfk2L4≤Ckfk·
B
−1
∞,∞
k∇fkL2. (6)
Testing (1a) byu|u|2and using (1c), we get 1
4 d
dtkuk4L4+k|u|∇uk2L2+1 2 ∇|u|2
2 L2
= Z
R3
(∇×ω)·u|u|2dx− Z
R3
(u·∇π)|u|2dx
= Z
R3
ω[∇×(u|u|2)]dx+ Z
R3
πu·∇|u|2dx (7) Testing (1b) byω|ω|2, and using (1c) infers that
1 4
d
dtkωk4L4+k|ω|∇·ωk2L2+1 2 ∇|ω|2
2 L2
= Z
R3
(∇×u)·ω|ω|2dx−2kωk4L4
= Z
R3
u[∇×(ω|ω|2)dx−2kωk4L4, (8)
4 Sadek Gala et al.
where we have used the following identities due to the divergence free property of the velocity fieldu:
Z
R3
(u·∇)u·u|u|2dx=0= Z
R3
(u·∇)ω·ω|ω|2dx.
Summing up (7) and (8), it follows that 1
4 d
dt(kuk4L4+kωk4L4) +k|u|∇uk2L2+1 2 ∇|u|2
2
L2+k|ω|∇·ωk2L2+1 2 ∇|ω|2
2 L2
= Z
R3
ω[∇×(u|u|2)]dx+ Z
R3
u[∇×(ω|ω|2)dx−2kωk4L4− Z
R3
(u·∇π)|u|2dx. (9) Using the H¨older inequality and the Young inequality and integrating by parts, we derive the estimate of the first three terms on the right-hand side of (9) as follows:
Z
R3
ω[∇×(u|u|2)]dx+ Z
R3
u[∇×(ω|ω|2)dx−2kωk4L4
≤ kukL4kωkL4(k|u||∇ukL2+k|ω||∇ωkL2)−2kωk4L4
≤ k|u||∇uk2L2+k|ω||∇ωk2L2+Ckuk4L4. (10) To estimate the last term of the right-hand side of (9), we have after integrating by parts and employing the H¨older inequality and the Young inequality,
Z
R3
πu·∇|u|2dx
≤ Z
R3
|π| |u|
∇|u|2 dx
≤Ckπk2L4kuk2L4+1 4 ∇|u|2
2 L2
≤Ckπk˙
B−1∞,∞k∇πkL2kuk2L4+1 4 ∇|u|2
2 L2
≤CkπkB˙−1
∞,∞kuk2L4k|u|∇ukL2+1 4 ∇|u|2
2 L2
≤Ckπk2B˙−1
∞,∞kuk4L4+1
2k|u||∇uk2L2+1 4 ∇|u|2
2 L2
, (11)
and hence, d
dt(ku(t)k4L4+kω(t)k4L4) +k|u|∇uk2L2+ ∇|u|2
2
L2+k|ω|∇·ωk2L2+k|ω||∇ωk2L2
≤C(1+kπk2˙
B−1∞,∞)kuk4L4.
Applying the Gronwall inequality,yields
ku(·,t)k4L4+kω(·,t)k4L4
≤(ku0k4L4+kω0k4L4)exp
C Z t
0
(1+kπ(·,τ)k2˙
B−1∞,∞)dτ
.
Taking the inner product of (1a) with −∆u, (1b) with −∆ ω inL2(R3), adding the resulting equations together, and using the Gagliardo-Nirenberg inequalities:
k∇ukL4≤Ckuk15
L4k∆uk45
L2 and k∇ukL2 ≤Ckuk12
L2k∆uk12
L2. we obtain,
1 2
d
dt(k∇u(·,t)k2L2+k∇ω(·,t)k2L2) +k∆uk2L2+k∆ ωk2L2+k∇∇·ωk2L2+2k∇ωk2L2
Z
R3
(u·∇)u·∆udx− Z
R3
(∇×ω)·∆udx+ Z
R3
(u·∇)ω·∆ ωdx− Z
R3
(∇×u)·∆ ωdx
≤ kukL4k∇ukL4k∆ukL2+k∇ωkL2k∆ukL2+kukL4k∇ωkL4k∆ ωkL2+k∇ukL2k∆ ωkL2
≤Ckuk65
L4k∆uk95
L2+kωk12
L2k∆ ωk12
L2k∆ukL2+kukL4kωk15
L4k∆ ωk95
L2+kuk12
L2k∆uk12
L2k∆ ωkL2
≤Ckuk12L4+Ckωk2L2+C(kuk12L4+kωk12L4) +Ckuk2L2+1
2(k∆uk2L2+k∆ ωk2L2)
≤C(1+kuk12L4+kωk12L4) +1
2(k∆uk2L2+k∆ ωk2L2).
This yields, d
dt(k∇u(·,t)k2L2+k∇ω(·,t)k2L2) +k∆uk2L2+k∆ ωk2L2+k∇∇·ωk2L2+2k∇ωk2L2
≤C(1+kuk12L4+kωk12L4).
Integrating the above inequality over(0,t), we have k∇u(t)k2L2+k∇ω(t)k2L2+
Z t 0
(k∆u(τ)k2L2+k∆ ω(τ)k2L2+k∇∇·ω(τ)k2L2
+2k∇ω(τ)k2L2)dτ
≤ k∇u0k2L2+k∇ω0k2L2+C Z t
0
(1+ku(τ)k12L4+kω(τ)k12L4)dτ. (12) On the other hand, combining a Sobolev embedding theorem ˙H1(R3),→L6(R3), (12) and (5), we obtain that
e+kπ(·,t)kL3 ≤e+Cku(·,t)k2L6≤e+Ck∇u(·,t)k2L2
≤e+C(k∇u0k2L2+k∇ω0k2L2) +C Z t
0
(1+ku(·,τ)k12L4)dτ
≤e+C(k∇u0k2L2+k∇ω0k2L2) +C(e+t)sup
0≤τ≤t
(1+ku(·,τ)k12L4)
≤C
e+k∇u0k2L2+k∇ω0k2L2
(e+t)sup
0≤τ≤t
(1+ku(·,τ)k12L4)
≤C0(e+t)exp
C Z t
0
(1+kπ(·,τ)k2˙
B−1∞,∞)dτ
, (13)
where the constantC0=C(e,k∇u0kL2,k∇ω0kL2,ku0kL4,kω0kL4). Using the fact that L3(R3)⊂B˙−1∞,∞(R3), it follows that
e+kπ(·,t)kL3≤C(e+t)exp
C Z t
0
(1+kπ(·,τ)k2˙
B−1∞,∞)dτ
(14) Now, taking the logarithm on both sides of (14), we can conclude that
log(e+kπ(·,t)kL3)≤log(C(e+t)) +C Z t
0
(1+kπ(·,τ)k2˙
B−1∞,∞)dτ. (15) For simplicity, set
Z(t) =log(e+kπ(·,t)kL3), E(t) =log(C(e+t)) +C
Z t 0
(1+k∇π(·,τ)k2˙
B−1∞,∞)dτ, (16)
6 Sadek Gala et al.
withE(0) =log(Ce). Then, the above inequality (15) implies that 0<Z(t)≤E(t)
from which we easily get
(e+Z(t))log(e+Z(t))≤(e+E(t))log(e+E(t)).
On the other hand, we have d
dtlog(e+E(t)) = 1 e+E(t)
1
e+t +C(1+k∇π(·,t)k2˙
B−1∞,∞)
≤ 1 e2+C
1+k∇π(·,t)k2˙
B−1∞,∞
e+E(t)
= 1 e2+C
1+k∇π(·,t)k2˙
B−1∞,∞
(e+E(t))ln(e+E(t))log(e+E(t))
≤ 1 e2+C
1+k∇π(·,t)k2˙
B−1∞,∞
(e+Z(t))ln(e+Z(t))log(e+E(t)) Applying the Gronwall inequality to log(e+E(t)), we find
log(e+E(t))
≤log(e+E(0))exp
T e2+C
Z t 0
1+k∇π(·,τ)k2˙
B−1∞,∞
(e+Z(τ))log(e+Z(τ))dτ
,
and equivalently
e+E(t)≤(e+E(0))
exp
T
e2+CR0t
1+k∇π(·,τ)k2 B˙−1 (e+Z(τ))log(e+Z∞,∞(τ))dτ
Using the fact thatL3(R3)⊂B˙−1∞,∞(R3), it follows from (16) that
t Z
0
kπ(·,τ)kB˙−1
∞,∞dτ≤C
t Z
0
kπ(·,τ)kL3dτ
≤(e+E(0))
exp
T
e2+C1R0t
1+kπ(·,τ)k2 B˙−1
∞,∞
(e+Z(τ))log(e+Z(τ))dτ
<∞. (17)
Hence by virtue of (12) and (17), we conclude that (u,ω)∈L∞ (0,T);H1(R3)
∩L2 (0,T);H2(R3) , which completes the proof of Theorem1.1.
u t There is no conflict of interest.
3 Acknowledgments
Part of the work was carried out while S. Gala was a long-term visitor at the University of Catania. The hospitality of Catania University is graciously acknowledged. The paper is partially supported by GNAMPA 2019. Research of M.A. Ragusa is partially supported by the Ministry of Education and Science of the Russian Federation (5-100 program of the Russian Ministry of Education). Research of M. Th´era is supported by the Australian Research Council (ARC) grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF.
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