# Logarithmically Improved Blow up Criterion for Smooths Solution to the 3D Micropolar Fluid Equations

Authors: Yin-Xia Wang; Hengjun Zhao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Blow-up criteria of smooth solutions for the 3D micropolar fluid equations are investigated. Logarithmically improved blow-up criteria are established in the Morrey-Campanto space.


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## Body

## 1. Introduction

This paper concerns the initial value problem for the micropolar fluid equations inℝ3(1.1)∂tu-(μ+χ)Δu+u⋅∇u+∇p-χ∇×w=0,∂tw-γΔw-κ∇∇⋅w+2χw+u⋅∇w-χ∇×u=0,∇⋅u=0
with the initial value(1.2)t=0:u=u0(x),w=w0(x),
where u(t,x), w(t,x), and p(t,x) stand for the velocity field, microrotation field, and the scalar pressure, respectively. And ν>0 is the Newtonian kinetic viscosity, κ>0 is the dynamics micro-rotation viscosity, and α,β,γ>0 are the angular viscosity (see, i.e., Lukaszewicz [1]).The micropolar fluid equations were first proposed by Eringen [2]. It is a type of fluids which exhibits the micro-rotational effects and micro-rotational inertia and can be viewed as a non-Newtonian fluid. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, for example, liquid crystals that are made up of dumbbell molecules, are of the same type. For more background, we refer to [1] and references therein. 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 research, and many interesting results have been obtained (see [3–8]). Regularity criterion of weak solutions to (1.1) and (1.2) in terms of the pressure was obtained (see [4]). Gala [5] established a Serrin-type regularity criterion for the weak solutions to (1.1) and (1.2) in Morrey-Campanato space. Wang and Chen [7] established the regularity criteria of weak solutions to (1.1) and (1.2) via the derivative of the velocity in one direction. A new logarithmically improved blow-up criterion of smooth solutions to (1.1) and (1.2) in an appropriate homogeneous Besov space is established by Wang and Yuan [8].Ifκ=0 and w=0, then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray [9] and Hopf [10] constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results are established (see [11–26]). Regularity criteria of weak solutions to the Navier-Stokes equations in Morrey space were obtained in [13, 21].The main aim of this paper is to establish two logarithmically blow-up criteria of smooth solution to (1.1), (1.2). Our results state as follows.Theorem 1.1.
Letu0,w0∈Hm(ℝ3)(m≥3) with ∇·u0=0. Assume that (u,w) is a smooth solution to (1.1) and (1.2) on [0, T). If u satisfies
(1.3)∫0T‖u(t)‖Ṁ2,3/r2/(1-r)1+ln⁡(e+‖u(t)‖L∞)dt<∞,0<r<1,
then the solution (u,w) can be extended beyond t=T.We have the following corollary immediately.Corollary 1.2.
Letu0,w0∈Hm(ℝ3)(m≥3) with ∇·u0=0. Assume that (u,w) is a smooth solution to (1.1) and (1.2) on [0, T). Suppose that T is the maximal existence time, then
(1.4)∫0T‖u(t)‖Ṁ2,3/r2/(1-r)1+ln⁡(e+‖u(t)‖L∞)dt=∞,0<r<1.Theorem 1.3.
Letu0,w0∈Hm(ℝ3)(m≥3) with ∇·u0=0. Assume that (u,w) is a smooth solution to (1.1) and (1.2) on [0, T). If u satisfies
(1.5)∫0T‖∇u(t)‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u(t)‖L∞)dt<∞,0<r<1,
then the solution (u,w) can be extended beyond t=T.One has the following corollary immediately.Corollary 1.4.
Letu0,w0∈Hm(ℝ3)(m≥3) with ∇·u0=0. Assume that (u,w) is a smooth solution to (1.1) and (1.2) on [0, T). Suppose that T is the maximal existence time, then
(1.6)∫0T‖∇u(t)‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u(t)‖L∞)dt=∞,0<r<1.The paper is organized as follows. We first state some important inequalities in Section2, which play an important role in the proof of our main result. Then, we prove the main result in Section 3 and Section 4, respectively.


## 2. Preliminaries

Firstly, we recall the definition and some properties of the space that we are going to use. The space plays an important role in studying the regularity of solutions to nonlinear differential equations.Definition 2.1.
For1<p≤q≤+∞, the Morrey-Campanato space Ṁp,q is defined by
(2.1)Ṁp,q={f∈Lloc⁡p(R3)∣‖f‖Ṁp,q=sup⁡x∈R3sup⁡R>0R3/q-3/p‖f‖Lp(B(x,R))<∞},
where B(x,R) denotes the ball of center x with radius R.
It is easy to verify thatṀp,q is a Banach space under the norm ∥·∥Ṁp,q. Furthermore, it is easy to check the following:
(2.2)‖f(λ⋅)‖Ṁp,q=λ-3/q‖f‖Ṁp,q,λ>0.
Morrey-Campanato spaces can be seen as a complement to Lp spaces. In fact, for p≤q, one has
(2.3)Lq=Ṁq,q⊂Ṁp,q.
one has the following comparison between Lorentz spaces and Morrey-Campanato spaces: for p≥2,
(2.4)L3/r(R3)⊂L3/r,∞(R3)⊂Ṁp,3/r(R3),
where Lp,∞ denotes the usual Lorentz (weak Lp) space.
In the proof of our main result, we need the following lemma which was given in [27].Lemma 2.2.
For0≤r<3/2, the space Żr is defined as the space of f(x)∈Lloc2(ℝ3) such that
(2.5)‖f‖Żr=sup⁡‖g‖Ḃ2,1r≤1‖fg‖L2<∞.
Then f∈Ṁ2,3/r if and only if f∈Żr with equivalence of norms. And the fact that
(2.6)L2⋂Ḣ1⊂Ḃ2,1r⊂Ḣr,0<r<1,
one has
(2.7)Ẋr⊂Ṁ2,3/r,
where Ẋr denotes the pointwise multiplier space from Ḣr to L2.We need the following lemma that is basically established in [28]. For completeness, the proof will be also sketched here.Lemma 2.3.
For0<r<1, the inequality
(2.8)‖f‖Ḃ2,1r≤C‖f‖L21-r‖∇f‖L2r
holds, where C is a positive constant that depends on r.Proof.
It follows from the definition of Besov spaces that(2.9)‖f‖Ḃ2,1r=∑i∈Z2ir‖Δif‖L2≤∑i≤j2ir‖Δif‖L2+∑i>j2i(r-1)2i‖Δif‖L2≤(∑i≤j22ir)1/2(∑i≤j‖Δif‖L22)1/2+(∑i≤j22i(r-1))1/2(∑i>j22i‖Δif‖L22)1/2≤C(2jr‖f‖L2+2j(r-1)‖f‖Ḣ1)=C(2jrA-r+2j(r-1)A1-r)‖f‖L21-r‖f‖Ḣ1r,
where A=(∥f∥Ḣ1)/(∥f∥L2). Choosing j such that 1/2≤2jrA-r≤1, from (2.9) we get
(2.10)‖f‖Ḃ2,1r≤(1+2j(r-1)A1-r)‖f‖L21-r‖f‖Ḣ1r≤C(1+(12)-1/r)‖f‖L21-r‖∇f‖L2r.
Therefore, we have completed the proof of Lemma 2.3.The following Lemma comes from [29].Lemma 2.4.
Assume that1<p<∞. For f,g∈Wm,p, and 1<q≤∞, 1<r<∞, one has
(2.11)‖∇α(fg)-f∇αg‖Lp≤C(‖∇f‖Lq1‖∇α-1g‖Lr1+‖g‖Lq2‖∇αf‖Lr2),
where 1≤α≤m and 1/p=1/q1+1/r1=1/q2+1/r2.In order to prove Theorem1.1, we need the following interpolation inequalities in three space dimensions.Lemma 2.5.
In three space dimensions, the following inequalities(2.12)‖∇f‖L4≤C‖f‖L21/8‖∇2f‖L27/8‖f‖L4≤C‖f‖L23/4‖∇3f‖L21/4‖∇2f‖L4≤C‖f‖L21/12‖∇3f‖L211/12‖∇2f‖L2≤C‖f‖L21/3‖∇3f‖L22/3
hold.


## 3. Proof of Theorem1.1

Proof.
Multiplying the first equation of (1.1) by u and integrating with respect to x over ℝ3, using integration by parts, we obtain
(3.1)12ddt‖u(t)‖L22+(μ+χ)‖∇u(t)‖L22=χ∫R3(∇×w)⋅udx
Similarly, we get
(3.2)12ddt‖w(t)‖L22+γ‖∇w(t)‖L22+κ‖∇⋅w‖L22+2χ‖w‖L22=χ∫R3(∇×u)⋅wdx.
Summing up (3.1) and (3.2), we deduce thats
(3.3)12ddt(‖u(t)‖L22+‖w(t)‖L22)+(μ+χ)‖∇u(t)‖L22+γ‖∇w(t)‖L22+κ‖∇⋅w‖L22+2χ‖w‖L22=χ∫R3(∇×w)⋅udx+χ∫R3(∇×u)⋅wdx.
Using integration by parts and Cauchy’s inequality, we obtain
(3.4)χ∫R3(∇×w)⋅udx+χ∫R3(∇×u)⋅wdx≤χ‖∇u‖L22+χ‖w‖L22.
Combining (3.3) and (3.4) yields
(3.5)12ddt(‖u(t)‖L22+‖w(t)‖L22)+μ‖∇u(t)‖L22+γ‖∇w(t)‖L22+κ‖∇⋅w‖L22+χ‖w‖L22≤0.
Integrating with respect to t, we have
(3.6)‖u(t)‖L22+‖w(t)‖L22+2∫0t(μ‖∇u(τ)‖L22+γ‖∇w(τ)‖L22)dτ+2κ∫0t‖∇⋅w(τ)‖L22dτ+2χ∫0t‖w(τ)‖L22dτ≤‖u0‖L22+‖w0‖L22.
Taking∇ to the first equation of (1.1), then multiplying the resulting equation by ∇u and using integration by parts, we obtain
(3.7)12ddt‖∇u(t)‖L22+(μ+χ)‖∇2u(t)‖L22=-∫R3∇(u⋅∇u)∇udx+χ∫R3∇(∇×w)∇udx.
Similarly, we get
(3.8)12ddt‖∇w(t)‖L22+γ‖∇2w(t)‖L22+κ‖∇⋅∇w‖L22+2χ‖∇w‖L22=-∫R3∇(u⋅∇w)⋅∇wdx+χ∫R3∇(∇×u)⋅∇wdx.
Combining (3.7) and (3.8) yields
(3.9)12ddt(‖∇u(t)‖L22+‖∇w(t)‖L22)+(μ+χ)‖∇2u(t)‖L22+γ‖∇2w(t)‖L22+κ‖∇∇⋅w‖L22+2χ‖∇w‖L22=-∫R3∇(u⋅∇u)∇udx+χ∫R3∇(∇×w)∇udx-∫R3∇(u⋅∇w)∇wdx+χ∫R3∇(∇×u)∇wdx.
Using integration by parts and Cauchy’s inequality, we obtain
(3.10)χ∫R3∇(∇×w)⋅∇udx+χ∫R3∇(∇×u)⋅∇wdx≤χ‖∇2u‖L22+χ‖∇w‖L22.
Using Hölder’s inequality, (2.8), and Young’s inequality, we obtain
(3.11)-∫R3∇(u⋅∇u)∇udx≤‖∇u‖L2‖∇u∇u‖L2≤C‖∇u‖Ṁ2,3/r‖∇u‖Ḃ2,1r‖∇u‖L2≤C‖∇u‖Ṁ2,3/r‖∇u‖Ḃ2,1r2-r‖∇2u‖L2r≤μ2‖∇2u(t)‖L22+C‖∇u‖Ṁ2,3/r2/(2-r)‖∇u‖L22.
Similarly, we have the following estimate:
(3.12)-∫R3∇(u⋅∇w)∇wdx≤‖∇w‖L2‖∇u∇w‖L2≤C‖∇u‖Ṁ2,3/r‖∇w‖Ḃ2,1r‖∇w‖L2≤C‖∇u‖Ṁ2,3/r‖∇w‖Ḃ2,1r2-r‖∇2w‖L2r≤γ2‖∇2w(t)‖L22+C‖∇u‖Ṁ2,3/r2/(2-r)‖∇w‖L22.
Combining (3.9)-(3.12) yields
(3.13)ddt(‖∇u(t)‖L22+‖∇w(t)‖L22)+μ‖∇2u(t)‖L22+γ‖∇2w‖L22+κ‖∇∇⋅w‖L22+χ‖∇w‖L22≤C‖∇u‖Ṁ2,3/r2/(2-r)(‖∇u‖L22+‖∇w‖L22)≤C‖∇u‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u‖L∞)(‖∇u‖L22+‖∇w‖L22)(1+ln⁡(e+‖∇u‖L∞))≤C‖∇u‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u‖L∞)(‖∇u‖L22+‖∇w‖L22)(1+ln⁡(e+‖∇3u‖L2+‖∇3w‖L2)),
where we have used
(3.14)H2(R3)↪L∞(R3).
For any T0≤t≤T, we set
(3.15)ϑ(t)=sup⁡T0≤τ≤t(‖∇3u(τ)‖L22+‖∇3w(τ)‖L22).
Thus, from (3.13), we have
(3.16)ddt(‖∇u(t)‖L22+‖∇w(t)‖L22)+μ‖∇2u(t)‖L22+γ‖∇2w‖L22+κ‖∇∇⋅w‖L22+χ‖∇w‖L22≤C‖∇u‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u‖L∞)(‖∇u‖L22+‖∇w‖L22)(1+ln⁡(e+ϑ(t))),∀T0≤t<T.
It follows from (3.8) and Gronwall’s inequality that
(3.17)‖∇u(t)‖L22+‖∇w(t)‖L22≤(‖∇u(T0)‖L22+‖∇w(T0)‖L22)exp⁡{C(1+ln⁡(e+ϑ(t)))∫T0t‖∇u(τ)‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u‖L∞)dτ}≤C0exp⁡{Cɛ[1+ln⁡(e+ϑ(t))]}≤C0exp⁡{2Cɛ[ln⁡(e+ϑ(t))]}≤C0(e+ϑ(t))2Cɛ,
provided that
(3.18)∫T0t‖∇u(τ)‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖∇u‖L∞)dτ<ɛ≪1,
where C0=∥∇u(T0)∥L22+∥∇w(T0)∥L22.
Applying∇m to the first equation of (1.1), then multiplying the resulting equation by ∇mu and using integration by parts, we have
(3.19)12ddt‖∇mu(t)‖L22+(μ+χ)‖∇m+1u(t)‖L22=-∫R3∇m(u⋅∇u)∇mudx+χ∫R3∇m(∇×w)∇mudx.
Likewise, from the second equation of (1.1), we obtain
(3.20)12ddt‖∇mw(t)‖L22+γ‖∇m+1w(t)‖L22+κ‖∇m∇⋅w‖L22+2χ‖∇mw(t)‖L22=-∫R3∇m(u⋅∇w)∇mwdx+χ∫R3∇m(∇×u)∇mwdx.
Using ∇·u=0 and (3.19) and (3.20), we have
(3.21)12ddt(‖∇mu(t)‖L22+‖∇mw(t)‖L22)+(μ+χ)‖∇m+1u(t)‖L22+γ‖∇m+1w(t)‖L22+κ‖∇m∇⋅w‖L22+2χ‖∇mw(t)‖L22=-∫R3[∇m(u⋅∇u)-u⋅∇∇mu]∇mudx+χ∫R3∇m(∇×w)∇mudx-∫R3[∇m(u⋅∇w)-u⋅∇∇mw]∇mwdx+χ∫R3∇m(∇×u)∇mwdx.
In what follows, for simplicity, we will set m=3.
By Hölder’s inequality, (2.11), (2.12), and Young’s inequality, we obtain
(3.22)-∫R3[∇3(u⋅∇u)-u⋅∇∇3u]∇3udx≤‖∇3(u⋅∇u)-u⋅∇∇3u‖L2‖∇3u‖L2≤C‖∇u‖L4‖∇3u‖L4‖∇3u‖L2≤C‖∇u‖L23/4‖∇4u‖L21/4‖∇u‖L21/12‖∇4u‖L211/12‖∇u‖L21/3‖∇4u‖L22/3≤C‖∇u‖L27/6‖∇4u‖L211/6≤μ4‖∇4u‖L22+C‖∇u‖L214≤μ4‖∇4u‖L22+C(e+ϑ(t))14Cɛ,(3.23)-∫R3[∇3(u⋅∇w)-u⋅∇∇3w]∇3wdx≤‖∇3(u⋅∇w)-u⋅∇∇3w‖L2‖∇3w‖L2≤C‖∇u‖L4‖∇3w‖L4‖∇3w‖L2+‖∇w‖L4‖∇3u‖L4‖∇3w‖L2≤C‖∇u‖L23/4‖∇4u‖L21/4‖∇w‖L21/12‖∇4w‖L211/12‖∇w‖L21/3‖∇4w‖L22/3+C‖∇w‖L23/4‖∇4w‖L21/4‖∇u‖L21/12‖∇4u‖L211/12‖∇w‖L21/3‖∇4w‖L22/3≤μ4‖∇4u‖L22+γ2‖∇4w‖L22+C‖∇u‖L29‖∇w‖L25+C‖∇u‖L2‖∇w‖L213≤μ4‖∇4u‖L22+γ2‖∇4w‖L22+C(e+ϑ(t))14Cɛ.
It follows from integration by parts and Cauchy’s inequality that
(3.24)χ∫R3∇3(∇×w)∇3udx+χ∫R3∇3(∇×u)∇3wdx≤χ‖∇4u(t)‖L22+χ‖∇3w(t)‖L22.
Combining (3.21)-(3.24) yields
(3.25)12ddt(‖∇mu(t)‖L22+‖∇mw(t)‖L22)+(μ+χ)‖∇m+1u(t)‖L22+γ‖∇m+1w(t)‖L22+κ‖∇m∇⋅w‖L22+2χ‖∇mw(t)‖L22≤C(e+ϑ(t))14Cɛ,∀T0≤t<T.
Taking ɛ small enough yields
(3.26)ddt(‖∇3u‖L22+‖∇3w‖L22)≤C(e+ϑ(t)),T0≤t<T,
for all T0≤t<T.
Integrating (3.26) with respect to time from T0 to τ, we have
(3.27)e+‖∇3u(τ)‖L22+‖∇3w(τ)‖L22≤e+‖∇3u(T0)‖L22+‖∇3w(T0)‖L22+C2∫T0τ(e+ϑ(s))ds.
Owing to (3.27), we get
(3.28)e+ϑ(t)≤e+‖∇3u(T0)‖L22+‖∇3w(T0)‖L22+C2∫T0t(e+ϑ(τ))dτ
For all T0≤t<T, with help of Gronwall inequality and (3.28), we have
(3.29)e+‖∇3u(t)‖L22+‖∇3w(t)‖L22≤C,
where C depends on ∥∇u(T0)∥L22+∥∇w(T0)∥L22. From (3.29) and (3.5), we know that (u,w)∈L∞(0,T;H3(ℝ3)). Thus, (u,w) can be extended smoothly beyond t=T. We have completed the proof of Theorem 1.1.


## 4. Proof of Theorem1.3

We start to estimate every term on the right of (3.9). Using integration by parts, Hölder inequality, (2.8) and Young inequality, we obtain(4.1)-∫R3∇(u⋅∇u)∇udx≤‖∇2u‖L2‖u∇u‖L2≤C‖u‖Ṁ2,3/r‖∇u‖Ḃ2,1r‖∇2u‖L2≤C‖u‖Ṁ2,3/r‖∇u‖Ḃ2,1r1-r‖∇2u‖L21+r≤μ2‖∇2u(t)‖L22+C‖u‖Ṁ2,3/r2/(1-r)‖∇u‖L22.
Similarly, we have the following estimate(4.2)-∫R3∇(u⋅∇w)∇wdx≤‖∇2w‖L2‖u∇w‖L2≤C‖u‖Ṁ2,3/r‖∇w‖Ḃ2,1r‖∇2w‖L2≤C‖u‖Ṁ2,3/r‖∇w‖Ḃ2,1r1-r‖∇2w‖L21+r≤γ2‖∇2w(t)‖L22+C‖u‖Ṁ2,3/r2/(1-r)‖∇w‖L22.
Thus from (3.9), (3.10), (4.1), and (4.2), we obtain(4.3)ddt(‖∇u(t)‖L22+‖∇w(t)‖L22)+μ‖∇2u(t)‖L22+γ‖∇2w‖L22+κ‖∇∇⋅w‖L22+χ‖∇w‖L22≤C‖u‖Ṁ2,3/r2/(1-r)1+ln⁡(e+‖u‖L∞)(‖∇u‖L22+‖∇w‖L22)(1+ln⁡(e+ϑ(t))),∀T0≤t<T.
It follows from (4.3) and Gronwall’s inequality that(4.4)‖∇u(t)‖L22+‖∇w(t)‖L22≤(‖∇u(T0)‖L22+‖∇w(T0)‖L22)exp⁡{C(1+ln⁡(e+ϑ(t)))∫T0t‖u(τ)‖Ṁ2,3/r2/(1-r)1+ln⁡(e+‖u‖L∞)dτ}≤C0exp⁡{Cɛ[1+ln⁡(e+ϑ(t))]}≤C0exp⁡{2Cɛ[ln⁡(e+ϑ(t))]}≤C0(e+ϑ(t))2Cɛ,
provided that(4.5)∫T0t‖u(τ)‖Ṁ2,3/r2/(2-r)1+ln⁡(e+‖u‖L∞)dτ<ɛ≪1,
where C0=∥∇u(T0)∥L22+∥∇w(T0)∥L22.From (4.4), Hm estimate for Theorem 1.3 is same as that for Theorem 1.1. Thus, Theorem 1.3 is proved.



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