Chin.Quart.J.of Math.
2020,35(4):418-423
Regularity Criterion of Weak Solutions to the3D Micropolar Fluid Equations in Besov Space
LI Xiao
(Department of Mathematics Teaching and Rearch,Huanghe Jiaotong University,Henan,454000,
China)
Abstract:This paper studies the regularity criterion of weak solutions to the micropolar
fluid equations in three dimensions.Let(V u e L2-y(0,T;B—Y』,V w e L2(0,T;^g),it
is showed that the weak solution(u,w)is globally regular for the ca0<y<2.
Keywords:Micropolar fluid equations;Regularity criteria;Besov space
2000MR Subject Classification:35Q35,35B65
CLC number:O175.29Document code:A
Article ID:1002-0462(2020)04-0418-06
DOI:10.13371力ki.chin.2020.04.010
§1.Introduction
In this paper we consider the regularity criterion of weak solutions for the incompressible micropolar fluid equations in terms of pressure in three dimensions:
dU—+u-V u+V p=0,
疇—y A z—KVdivz+2x3+u-V®—%V x u=0,
divu=0,
、u(x,0)=u o(x),3(x,0)=3o(x),
where u=(u i(x,t),u2(x,t),u s(x,t))denotes the velocity of the fluid at a point x e R3and time t e[0,T);3=(3i(x,t),32(x,t),33(x,t))and p=p(x,t)stand for the micro-rotational velocity and the hydrostatic pressure,
respectively,u°and3°are the prescribed initial data for the velocity and angular velocity with property divu°=0，卩is the kinematic viscosity,x is the vortex viscosity,k and y are spin viscosities.
A theory of Micropolar fluid system was firstly developed by Eringen[6].It is a type of fluids which exhibits micro-rotational effects and micro-rotational inertia,and can be viewed as a non-Newtonian fluid.Physically,micropolar fluids reprent fluids that consist of rigid,randomly oriented(or spherical particles)suspended in a viscous medium,where the deformation of fluid
Received date:2020-04-10
Biographies:LI Xiao(1991-),femal,native of Fanxian,Henan,graduate student,engages in partial differential equations and applications.
418
No.4LI Xiao:Regularity Criterion of Weak Solutions---419
particles is ignored.It can describe many phenomena that appear in a large number of complex fluids such as the suspensions,animal blood,liquid crystals which cannot be characterized appropriately by the Navier-Stokes system.For more background,we refer to[12]and references therein.
Mathematically,Galdi and Rionero considered the weak solutions in[7].Using linearization and an almost fixed point theorem,Lukaszewicz[13]established the global existence of weak solutions with sufficiently regular initial data.And using the same technique,Lukaszewicz[14] proved the local and global existence and the uniqueness of the strong solutions under asymmetric condition.Yamaguchi[19]proved the existence theorem of global in time solution for small initial data.
When the micro-rotation effects are neglected or•=0,the micropolar fluid flows reduces to the incompressible Navier-Stokes equations,which has been greatly analyzed,e,for example, the classical books by Ladyzhenskaya[9],Lions[11]or Lemari6-Rieust[10].From the viewpoint of the model,therefore,Navier-Stokes flow is viewed as the flow of a simplified micropolar fluid.
Besides their physical applications,the micropolar fluid equations are also mathematically significant.Fundamental mathematical issues such as the global regularity of their solutions have generated extensive rearch,and many interesting results have been obtained(e,for example,[3,4,8,12,15,18]and references therein).
The regularity of weak solutions and blow-up criteria of smooth solutions to the micropolar fluid equations are important topic in the rearch of global well-podness.Yuan[21]established classical Serrin-type regularity criteria in terms of the velocity or its gradient
辣白菜五花肉炒饭
23
u w L q(0,T;L p)(R3),—+—=1,3<p Sx
q P
and
233
绿豆甜汤V u w L q(0,T;L p)(R3),-+-=2,-<P<X.
q p2
Particularly,in the end-point ca p=x,the blow-up criteria can be extended to more general spaces Vu w L x(0,T;B±m).Later Yuan[20]extended the Serrin's regularity criteria to Lorentz spaces,and Gala[8]extended the Serrin's regularity criteria to the Morrey-Campanto spaces.
Dong[5]further refined the velocity regularity in general Besov spaces
u W L皋(0,T;B^i TO)(R3),-1<r<1.
Recently,two new logarithmically blow-up criteria of smooth solution to the equations (1.1)in the Morrey-Campanto space are established by Wang and Zhao[18],and Zhang[17] established an improved blow-up criteria in terms of vorticity of velocity in Besov
严||V x训B o
/f二dt<x.
J0A/1+log(1+|Vx u\I jb o)
420CHINESE QUARTERLY JOURNAL OF MATHEMATICS Vol. 35
Motivated by the reference mentioned above, the purpo of the prent paper is to extend the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations (1.1) in terms of partial derivatives of the velocity and the micro-rotational velocity.
Before stating our main results we introduce some function spaces and notations. Let (R 3) denote the t of all C m vector functions f (x) = (/i (x),/2(x),/3(x)) with compact support such that divf (x) = 0. L ； (R 3) is the closure of C Q ^a (R 3)-function with respect to the L r -norm
|| - ||r for 1 <r <x . H S (R 3) denotes the closure of C (^a (R 3) with respect to the H s -norm ||f ||炉=||(1 —△)2f H 2, for s >0.
Now, we state our results as follows.
Theorem 1.1. Let u q W H 1 (R 3) and 3° W H X (R 3). Suppo that (u(t,x),^(t,x)) is a weak solution to the equations (1.1) and satisfies the strong energy inequality. If (u,3)satisfies
r T 2
(|V u(t)||B 二 + ||V3(t)||B
— i )dT< X ,
then the weak solution (u,3)is regular on (0,T ].Next, in order to derive the criteria on regularity of weak solutions to the micropolar fluid equations (1.1), we introduce the definition of a weak solution.Definition 1.1. Let u °(x) W L ^(R 3) and 3°(x) W L 2 (R 3). A measurable function (u(x,t),3(x,t)) is called a weak solution to the micropolar equations (1.1) on [0,T ] if
(a)
u(x,t) W L TO (0,T ;L 2(R 3)) n L 2(0,T ;H l (R 3))
and
(b)
3 W L TO (0,T ;L 2(R 3)) n L 2(0,T ;H X (R 3));
{-(u,d T 0)+ (“ + x)(V u, V ^) + (u ・V u,0)}— x(V x 3,^)dr
禾麻
=—(u 0,^(0)),
{ —(3，篦 0)+ y (V 3, V 0) + K (div3,div0) + 2x (3,0) + (u - V 3,0) — x(V x u,0)dT = -(30,0(0)),
for any 卩(x,t) W H 1([0,T ]; H 1 (R 3) and 0(x,t) W H 1([0,T ]; H 1(R 3) with 卩(T)=0 and 0(T) = 0. In the reference [16], Rojas-Medar and Boldrini proved the global existence of weak solutions to the equations (1.1) of the magneto-micropolar fluid motion by the Galerkin method. The weak solutions also satisfy the strong energy inequality
IIVulRds + 2了/ |V3|2ds + 2k / ||div3||2
ds
No. 4LI Xiao: Regularity Criterion of Weak Solutions---421
Il 3||2ds <\\(u o ,3o )\\l ,
Lemma 1.1. (page 82 in [1]) Let 1 < q <p < x and a be a position real number. A constant C exists such that
with 0 = a(P — 1) and 0 =qp In particular, for q = 2, p = 3,we have
休闲裤男装
1 2\f \L3 < C Ilf ||B 「Ilf
2 .
(1.2)
Another situation, for 0 = 1, q = 2 and p = 4 , we get a = 1 and 11If IL < C Ilf ||B -」|f 賂 1.
(1.3)
Then we prove Theorem 1.1 in Section 2 as follows.§2. Proof of Theorem 1・1
Taking the inner product of —Au with the first equation of (1.1), and taking the inner product of —A® with the cond equation of (1.1). By integrating by parts and using the incompressibility condition, we have
(|V u |2 + ll V3\l ) + (" + x)\ △训 1 + 7|A3\i + K \div V 3\2
x / V x V 3 - V udx + x / V x V u - V ®dx — 2x||V3||i J r 3 J r 3<2xii V 3iii +211 △训1—2x IV 3I i =211 △训1.
Substituting (2.2) into (2.1) yields that
(IVuI i + ll V3I i ) +(" + 学)II △训1 + y I A3I 1 + K ||divV3||2
< V u - V u - V udx + V u - V ® - V ®dx.
7r 3 7r 3<X / V x V 3 - V udx + x/ V x V u - V ®dx — 2XII V 3I 1
梦见别人杀我
J r 3 J r 3(2.1)
▽u • • \/3dx.
for 0 < s < t < T .
B x , x q,
Employing the Holder and Young inequalities and integration by parts, we derive the estimation of the first three terms on the right-hand side of (2.1) as
(2.2)
(2.3)
422CHINESE QUARTERLY JOURNAL OF MATHEMATICS Vol. 35Next we estimate the estimation of the first two terms on the right-hand side of (2.3). By means of the Holder and Young inequalities, as well as (1.2), we have
V u - V u - V udx <||Vu||3
<C I V u I B Xx IWG Y <C ||Vu||B X Y x ||Vu|「||Vu||；
<C||Vu|負 ||Vu||i + 豊||Au||i ,
B CO , CO 2
where we have ud the interpolation inequality
2 2 — y y IV 训H 2 <C||V 训2 Y ||V u||；.
Arguing the Hoolder and Young inequalities and combing the inequality (1.3), one has
/ Vu -V ®-V ®dx<||Vu||i||V 3||4
丿r 3
<C ||V 训 i||V3||B x 1x I A 3||i <C||V3||B-1
||Vu||i + 2 ||A3||i .B B x , CO 2Inrting (2.2), (2.4) and (2.5) into (2.1), we find
脚的英文复数
(I ▽训1 + ll V3I i ) + (“；％)”△训i + 2 ll A3I i + K ||div V 3I i
2<C(||Vu||i + ||V3||i )(||Vu||B 二 + ||V3||B -1 ).B x ,x B x ,x Then, by means of Gronwall inequality and energy inequality (1.2), we finally have
IIV u II 1 + ||V3||i
+ (“ + x) / ||Au||i dT+ / y I|A3||1 dr + 2^ IdivVz^dT o o o
<(IV u o (x )||i + ||V 3o (x )||i + C ||V ”u o (x )|| 訂 exp (||V 训 + ||V 3||B x J d (2.4)
(2.5)
(2.6)
(2.7)<x .
Thus, we complete the proof of Theorem 1.1.
[References]
[1] BAHOURI H, DANCHIN R, CHEMIN J Y. Fourier analysis and nonlinear partial differential equations[M]. Berlin Heidelberg: Springer, 2011.
[2] BERGH J, LOFSTROM J L. Inerpolation spaces, an introduction[M]. New York: Springer, 1976.
[3] CHEN Q L, MIAO C X. Global well-podness for the micropolar fluid system in critical Besov spaces[J]. J. Differ. Equ., 2012, 252: 2698-2724.
[4] DONG B Q, CHEN Z M. Regularity criteria of weak solutions to the three-dimensional micropolar flows[J]. J. Math. Phys., 2009, 50: 1-13.
[5] DONG B Q, ZHANG W. On the regularity criterion for the three-dimensional micropolar flows in Besov spaces[J]. Nonlinear Anal., 2010, 73:
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