# Blow-Up Criteria of Smooth Solutions for the Cahn-Hilliard-Boussinesq System with Zero Viscosity in a Bounded Domain

Authors: Yong Zhou; Jishan Fan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.


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## Body

## 1. Introduction

LetΩ⊆ℝ3 be a bounded, simply connected domain with smooth boundary ∂Ω, and n is the unit outward normal vector to ∂Ω. We consider the following Cahn-Hilliard-Boussinesq system with zero viscosity in Ω×(0,∞) [1]:(1.1)∂tu+(u⋅∇)u+∇π=μ∇ϕ+θe3,(1.2)div⁡u=0,(1.3)∂tθ+u⋅∇θ=Δθ,(1.4)∂tϕ+u⋅∇ϕ=Δμ,(1.5)-Δϕ+f′(ϕ)=μ,(1.6)u⋅n=0,∂θ∂n=∂ϕ∂n=∂μ∂n=0on⁡∂Ω×(0,∞),(1.7)(u,θ,ϕ)(x,0)=(u0,θ0,ϕ0)(x),x∈Ω,
where u, the fluid velocity field, θ, the temperature, ϕ, the order parameter, π, and the pressure are the unknowns. e3:=(0,0,1)t. μ is a chemical potential. f(ϕ):=(1/4)(ϕ2-1)2 is the double well potential.Whenθ=ϕ≡0, (1.1) and (1.2) are Euler equations. Ogawa-Taniuchi [3] proved that a smooth solution breaks down if a certain norm of vorticity blows up at the same time. Here this norm is weaker than bmo-norm.Before presenting our results, we introduce some function spaces and some notations.Letη,ϕj,j=0,±1,±2,±3,… be the Littlewood-Paley dyadic decomposition of unity that satisfies(1.8)η∈C0∞(B(0,1)),ϕ∈C0∞(B(0,2)∖B(0,12)),ϕj(ξ)=ϕ(2-jξ),η(ξ)+∑j=0∞ϕj(ξ)=1,
for any ξ∈ℝ3, where B(x,r) denotes the ball centered at x of radius r.We first recall the space of Besov type introduced by Vishik [2].Definition 1.1 (see [2]).
LetΘ(α)(≥1) be a nondecreasing function on [1,∞). VΘ:={f∈𝒮′;∥f∥VΘ<∞} is introduced by the norm
(1.9)‖f‖VΘ:=sup⁡N=1,2,…⁡‖(ηf̂)∨‖L∞+∑j=0N‖(ϕjf̂)∨‖L∞Θ(N),
where f̂ and f̌ denote the Fourier and inverse Fourier transforms.We note that(1.10)‖f‖VΘ≤C‖f‖Ḃ∞,∞0≤C‖f‖bmo≤C‖f‖L∞,ifΘ(N)≥N.Now let us introduce the space of bmo type in [3].Definition 1.2.
Letβ(r) be a positive function on (0,1], and Ω⊂ℝ3 is a domain with ∂Ω∈C∞.(1)
bmoβ(ℝ3) is the space defined as a set for an Lloc1(ℝ3) function f such that
(1.11)‖f‖bmoβ(R3)∶=sup⁡0<r<1,x∈R31|B(x,r)|β(r)∫B(x,r)|f(y)-f¯B(x,r)|dy+sup⁡x∈R31|B(x,1)|∫B(x,1)|f(y)|dy<∞,
where f¯B:=(1/|B|)∫Bf(y)dy.(2)
OnΩ⊂ℝ3 we define bmoβ as restrictions of the above space bmoβ(ℝ3):(1.12)bmoβ(Ω):={f∣Ω;f∈bmoβ(R3)},
where f|Ω is the restriction of f on Ω. The norm of this space is defined by
(1.13)‖f‖bmoβ(Ω):=inf⁡{‖f̃‖bmoβ(R3);f̃∈bmoβ(R3)withf̃=finΩ}.In particular ifβ(r)=1, we write bmoβ(ℝ3)=bmo(ℝ3) and bmoβ(Ω)=bmo(Ω). Obviously, bmo⊂bmoβ if β≥1.Definition 1.3.
LetΘ(α)(≥1) be a nondecreasing function on [1,∞)(1.14)YΘ(Ω):={f∈L1(Ω);‖f‖YΘ(Ω)<∞},
where
(1.15)‖f‖YΘ(Ω):=sup⁡p≥1‖f‖LpΘ(p),MΘ(Ω):={f∈L1(Ω);‖f‖MΘ(Ω)<∞},
where
(1.16)‖f‖MΘ(Ω):=sup⁡p≥11Θ(p)sup⁡0<r<1,x∈R3(r-3+3/p∫B(x,r)∩Ω|f(y)|dy).We note that these spaces have the following relations:(1.17)‖f‖MΘ(Ω)≤C‖f‖YΘ(Ω)≤C‖f‖bmo(Ω).From now on we impose the following assumptions.Assumption 1.4.
Letβ(r):=Θ(log(e+1/r))/log(e+1/r).(H1)
Θ(α) is a positive and nondecreasing function on [0,∞) satisfying(1.18)∫+∞dαΘ(α)=∞,Θ(α)≥α.(H2)
For alls≥1 there exists C(s) such that(1.19)Θ(sα)≤C(s)Θ(α),∀α≥1.(H3)
β(r) is a nonincreasing function on (0,1].
Then Ogawa-Taniuchi [3] proved the following blowup criterion:
(1.20)∫0T‖ω(t)‖bmoβ(Ω)+‖ω(t)‖MΘ(Ωϵ)dt=∞,
where ω:=curlu and for all ϵ>0 and Ωϵ:={x∈Ω;dist(x,∂Ω)<ϵ} or
(1.21)∫0T‖ω(t)‖bmoβ(Ω3ϵ)+‖ω(t)‖MΘ(Ω3ϵ)+‖ρω(t)‖VΘdt=∞,
for all 0<ϵ<ϵ0 and all ρ∈C∞(ℝ3) with ρ≡1 inΩ∖Ωϵ and ρ≡0 in ℝ3∖Ω. ϵ0 is a small positive constant depending only on Ω.
Sinceβ(r)≥1, we see
(1.22)‖f‖bmoβ(Ω)≤‖f‖bmo(Ω).
By this inequality and (1.17), (1.20) implies
(1.23)∫0T‖ω(t)‖bmo(Ω)dt=∞.
The aim of this paper is to prove a similar result for the problem (1.1)–(1.7). It is easy to show that the problem (1.1)–(1.7) has a unique local smooth solution. Thus, we omit the details here. However, the global regularity is still open, which this paper aims to study. We will prove that.Theorem 1.5.
Letu0∈H3,θ0∈H2,ϕ0∈H4,divu0=0 in Ω, u0·n=∂θ0/∂n=∂ϕ0/∂n=∂μ0/∂n=0 on ∂Ω. Suppose that (u,θ,ϕ) is a local smooth solution to the problem (1.1)–(1.7) on [0,T). If T is maximal, then (1.20) and (1.21) hold true.In Section2, we will give some preliminaries. Section 3 is devoted to the proof of Theorem 1.5.


## 2. Preliminaries

Lemma 2.1 (see [4]).
For anyu∈Ws,p with divu=0 in Ω and u·n=0 on ∂Ω, there holds
(2.1)‖u‖Ws,p≤C(‖u‖Lp+‖curl⁡u‖Ws-1,p),
for any s≥1 and p∈(1,∞).Lemma 2.2 (see [5]).
Lets≥1.(1)
Iff,g∈Hs(Ω)∩C(Ω), then
(2.2)‖fg‖Hs(Ω)≤C(‖f‖Hs(Ω)‖g‖L∞(Ω)+‖f‖L∞(Ω)‖g‖Hs(Ω)).(2)
Iff∈Hs(Ω)∩C1(Ω) and g∈Hs-1(Ω)∩C(Ω), then for |α|≤s,
(2.3)‖Dα(fg)-fDαg‖L2(Ω)≤C(‖f‖Hs(Ω)‖g‖L∞(Ω)+‖f‖W1,∞(Ω)‖g‖Hs-1(Ω)).Lemma 2.3 (see [3]).
For allϵ>0, there holds
(2.4)‖∇u‖L∞(Ω)≤C(1+‖u‖L2(Ω)+‖curl⁡u‖bmoβ(Ω)+‖curl⁡u‖MΘ(Ωϵ))×Θ(log⁡(e+‖u‖H3(Ω))),
for all u∈H3(Ω) with divu=0 in Ω and u·n=0 on ∂Ω.Lemma 2.4 (see [3]).
There exists a constantϵ0 depending only on Ω such that the following holds.
For all0<ϵ<ϵ0, and for all ρ∈C∞(ℝ3) with ρ≡1 in Ω∖Ωϵ and ρ≡0 in ℝ3∖Ω, there exists constant C depending only on ϵ,ρ,Ω and Θ such that
(2.5)‖∇u‖L∞(Ω)≤C(1+‖u‖L2(Ω)+‖curl⁡u‖bmoβ(Ω3ϵ)+‖curl⁡u‖MΘ(Ω3ϵ)+‖ρcurl⁡u‖VΘ‖curl⁡u‖bmoβ(Ω3ϵ))Θ(log⁡(e+‖u‖H3(Ω))),
for all u∈H3(Ω) with divu=0 in Ω and u·n=0 on ∂Ω.Lemma 2.5 (see [6]).
Letψ be nonnegative function on (0,T) with ∫0Tψ(t)dt<∞, let Θ(α) be a positive and nondecreasing for α≥1 and ∫+∞(dα/Θ(α))=∞. Assume that v∈C([0,T)) and
(2.6)0≤v(t)≤v(0)+∫0tψ(s)Θ(v(s))ds∀0≤t<T.
Then, sup0≤t≤Tv(t)<∞.


## 3. Proof of Theorem1.5

In this section, all the integrations with respect to spacial variable are on the domainω (we omit it for simplicity).Since the proof of (1.21) is similar to that of (1.20), we only need to prove (1.20). By the standard argument of continuation of local solutions, it suffices to prove that if(3.1)∫0T‖ω(t)‖bmoβ(Ω)+‖ω(t)‖MΘ(Ωϵ)dt<∞forsomeϵ>0,
then(3.2)u∈L∞(0,T;H3),θ∈L∞(0,T;H2)∩L2(0,T;H3),ϕ∈L∞(0,T;H4).First, by the maximum principle, it follows from (1.2) and (1.3) that(3.3)‖θ‖L∞(0,T;L∞)≤C.Testing (1.3) by θ, using (1.2), we see that(3.4)12ddt∫θ2dx+∫|∇θ|2dx=0,
whence(3.5)‖θ‖L2(0,T;H1)≤C.Testing (1.4) by ϕ, using (1.2) and (1.6), we find that(3.6)12ddt∫ϕ2dx+∫|Δϕ|2dx=∫f′(ϕ)Δϕdx=∫(ϕ3-ϕ)Δϕdx=-3∫ϕ2|∇ϕ|2dx-∫ϕΔϕdx≤-∫ϕΔϕdx≤‖ϕ‖L2‖Δϕ‖L2≤12‖Δϕ‖L22+12‖ϕ‖L22,
which gives(3.7)‖ϕ‖L∞(0,T;L2)+‖ϕ‖L2(0,T;H2)≤C.Testing (1.1) and (1.4) by u and μ, respectively, using (1.2), (1.5), (1.6), and (3.3), we infer that
(3.8)ddt∫12|∇ϕ|2+f(ϕ)+12u2dx+∫|∇μ|2dx=∫θe3udx≤‖θ‖L2‖u‖L2≤C‖u‖L2,
which yields(3.9)‖ϕ‖L∞(0,T;H1)≤C,(3.10)‖u‖L∞(0,T;L2)≤C,(3.11)‖∇μ‖L2(0,T;L2)≤C.In the following calculations, we will use the following Gagliardo-Nirenberg inequality:(3.12)‖ϕ‖L∞2≤C‖ϕ‖H1‖ϕ‖H2.It follows from (1.5), (3.11), (3.7), (3.9) and (3.12) that(3.13)∫0T∫|∇Δϕ|2dxdt=∫0T∫|∇(f′(ϕ)-μ)|2dxdt≤C∫0T∫|∇μ|2dxdt+C∫0T∫|∇f′(ϕ)|2dxdt≤C+C∫0T∫|∇(ϕ3-ϕ)|2dxdt≤C+C∫0T∫ϕ2|∇ϕ|2dxdt≤C+C‖∇ϕ‖L∞(0,T;L2)2∫0T‖ϕ‖L∞4dt≤C+C∫0T‖ϕ‖L∞4dt≤C+C∫0T‖ϕ‖H12‖ϕ‖H22dt≤C+C‖ϕ‖L∞(0,T;H1)2∫0T‖ϕ‖H22dt≤C,
which implies
(3.14)‖∇ϕ‖L2(0,T;L∞)≤C,‖ϕ‖L4(0,T;L∞)≤C.Testing (1.4) by Δ2ϕ, using (3.9), (3.10), and (3.14), we deduce that(3.15)12ddt∫|Δϕ|2dx+∫|Δ2ϕ|2dx=∫Δf′(ϕ)⋅Δ2ϕdx-∫(u⋅∇)ϕ⋅Δ2ϕdx≤(‖Δf′(ϕ)‖L2+‖u‖L2‖∇ϕ‖L∞)‖Δ2ϕ‖L2≤C(‖ϕ‖L∞2‖Δϕ‖L2+‖ϕ‖L∞‖∇ϕ‖L∞‖∇ϕ‖L2+‖∇ϕ‖L∞)‖Δ2ϕ‖L2≤12‖Δ2ϕ‖L22+C‖ϕ‖L∞4‖Δϕ‖L22+C‖ϕ‖H22‖∇ϕ‖L∞2+C‖∇ϕ‖L∞2,
which leads to(3.16)‖ϕ‖L∞(0,T;H2)+‖ϕ‖L2(0,T;H4)≤C.
Testing (1.3) by -Δθ, using (1.2) and (1.6), we infer that(3.17)12ddt∫|∇θ|2dx+∫(Δθ)2dx=∫(u⋅∇)θ⋅Δθdx=∑i∫∂i((u⋅∇θ)∂iθ)dx-∑i∫∂i(u⋅∇θ)⋅∂iθdx=∫div⁡((u⋅∇θ)∇θ)dx-∑i∫∂iu⋅∇θ⋅∂iθdx-12∑i∫u⋅∇(∂iθ)2dx=-∑i∫∂iu⋅∇θ⋅∂iθdx≤C‖∇u‖L∞‖∇θ‖L22.
Equations (1.3) and (1.6) can be rewritten as(3.18)Δθ=g:=∂tθ+u⋅∇θasΩ×(0,∞),∂θ∂n=0,on⁡∂Ω×(0,∞).By the classical regularity theory of elliptic equation, using (3.10), we get(3.19)‖θ‖H3≤C‖g‖H1≤C‖∂tθ‖H1+C‖u⋅∇θ‖H1≤C‖∂tθ‖H1+C‖u‖L2‖∇θ‖L∞+C‖u‖L6‖Δθ‖L3+C‖∇u‖L∞‖∇θ‖L2≤C‖∂tθ‖H1+C‖∇θ‖L∞+C‖u‖L6‖Δθ‖L3+C‖∇u‖L∞‖∇θ‖L2.Now using the following Gagliardo-Nirenberg inequalities:(3.20)‖∇θ‖L∞≤C‖θ‖L∞1/3‖θ‖H32/3,‖Δθ‖L3≤C‖θ‖L∞1/3‖θ‖H32/3,‖u‖L63≤C‖u‖L22‖u‖H3,
we obtain(3.21)‖θ‖H3≤C‖∂tθ‖H1+C+C‖u‖H3+C‖∇u‖L∞‖∇θ‖L2.Takingcurlto (1.1), using (1.2) and (1.5), we have(3.22)∂tω+u⋅∇ω=ω⋅∇u-curl⁡(Δϕ∇ϕ)+curl⁡(θe3).TakingΔ to (3.22), testing by Δω, using (1.2) and (1.6), we derive(3.23)12ddt∫|Δω|2dx=-∫(Δ(u⋅∇ω)-u∇Δω)Δωdx+∫Δ(ω⋅∇u)⋅Δωdx-∫Δcurl⁡(Δϕ⋅∇ϕ)⋅Δωdx+∫Δcurl⁡(θe3)⋅Δωdx≤(‖Δ(u⋅∇ω)-u∇Δω‖L2+‖Δ(ω⋅∇u)‖L2+‖Δcurl⁡(Δϕ∇ϕ)‖L2+‖Δcurl⁡(θe3)‖L2)‖Δω‖L2∶=(I1+I2+I3+I4)‖Δω‖L2.Using (1.2), (1.6), Lemma 2.2, I1 and I2 can be bounded as follows: (3.24)I1=∑i‖Δ∂i(uiω)-ui∂iΔω‖L2≤C‖∇u‖L∞‖Δω‖L2+C‖ω‖L∞‖∇3u‖L2≤C‖∇u‖L∞‖u‖H3,I2≤C‖ω‖L∞‖u‖H3+C‖∇u‖L∞‖ω‖H2≤C‖∇u‖L∞‖u‖H3.Using Lemma2.2, I3 can be bonded as follows(3.25)I3=‖∑jΔcurl⁡∂j(∂jϕ∇ϕ)‖L2≤C‖∇ϕ‖L∞‖∇ϕ‖H4≤C‖∇ϕ‖L∞‖ϕ‖H5.Inserting the above estimates into (3.23), we obtain(3.26)12ddt∫|Δω|2dx≤C(‖∇u‖L∞‖u‖H3+‖∇ϕ‖L∞‖ϕ‖H5+‖θ‖H3)‖Δω‖L2≤C‖∇u‖L∞‖u‖H33+C‖∇ϕ‖L∞2‖Δω‖L22+C‖Δω‖L22+δ‖ϕ‖H52+δ‖θ‖H32,
for any 0<δ<1.Testing (1.1) by ∂tu, using (1.2), (1.6), (3.3), (3.10), and (3.16), and noting that(3.27)∫(μ⋅∇)ϕ⋅∂tudx=-∫Δϕ∇ϕ⋅∂tudx,
we reach(3.28)‖∂tu‖L2≤‖θ‖L2+‖u⋅∇u‖L2+‖Δϕ∇ϕ‖L2≤C+C‖u‖L6‖∇u‖L3+C‖Δϕ‖L2‖∇ϕ‖L∞≤C+C‖u‖L22/3‖u‖H31/3⋅‖u‖L21/2‖u‖H31/2+C‖∇ϕ‖L∞≤C+C‖u‖H35/6+C‖∇ϕ‖L∞.
Here, we have used the Gagliardo-Nirenberg inequality:(3.29)‖∇u‖L32≤C‖u‖L2‖u‖H3.Taking∂t to (1.3), we see that(3.30)∂t2θ+u⋅∇∂tθ-Δ∂tθ=-∂tu⋅∇θ.Testing the above equation by∂tθ, using (1.2), (1.6) and (3.3), we have(3.31)12ddt∫|∂tθ|2dx+∫|∇∂tθ|2dx=-∫∂tu⋅∇θ⋅∂tθdx=∫∂tu⋅θ∇∂tθdx≤‖θ‖L∞‖∂tu‖L2‖∇∂tθ‖L2≤C‖∂tu‖L2‖∇∂tθ‖L2≤12‖∇∂tθ‖L22+C‖∂tu‖L22,
whence(3.32)ddt∫|∂tθ|2dx+∫|∇∂tθ|2dx≤C+C‖u‖H32+C‖∇ϕ‖L∞2.By the classical regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (3.16), and (3.10) that(3.33)‖ϕ‖H5≤C‖f′(ϕ)-μ‖H3≤C‖f′(ϕ)‖H3+C‖μ‖H3≤C‖f′(ϕ)‖H3+C‖∂tϕ+u⋅∇ϕ‖H1≤C+C‖ϕ‖H3+C‖∂tϕ‖H1+C‖∇u‖L∞‖∇ϕ‖L2+C‖u‖L2‖Δϕ‖L∞≤C+C‖ϕ‖H4+C‖∂tϕ‖H1+C‖u‖H3≤C+C‖ϕ‖H21/3‖ϕ‖H52/3+C‖∂tϕ‖H1+C‖u‖H3,
which implies(3.34)‖ϕ‖H5≤C+C‖∂tϕ‖H1+C‖u‖H3.
Here, we have used the Gagliardo-Nirenberg inequality:(3.35)‖ϕ‖H4≤C‖ϕ‖H21/3‖ϕ‖H52/3.Taking∂t to (1.4) and (1.5), we have(3.36)∂t2ϕ+u⋅∇∂tϕ=Δ∂tμ-∂tu⋅∇ϕ,(3.37)-Δ∂tϕ+∂t(ϕ3-ϕ)=∂tμ.Testing (3.36) by ∂tϕ, using (3.37), (1.4), (1.6), (3.16), and (3.28), we arrive at(3.38)12ddt∫|∂tϕ|2dx+∫|Δ∂tϕ|2dx=∫∂t(ϕ3-ϕ)⋅Δ∂tϕdx-∫∂tu⋅∇ϕ⋅∂tϕdx=∫(3ϕ2-1)∂tϕ⋅Δ∂tϕdx+∫∂tu⋅ϕ∇∂tϕdx≤‖3ϕ2-1‖L∞‖∂tϕ‖L2‖Δ∂tϕ‖L2+‖∂tu‖L2‖ϕ‖L∞‖∇∂tϕ‖L2≤C‖∂tϕ‖L2‖Δ∂tϕ‖L2+C‖∂tu‖L2‖∇∂tϕ‖L2≤C‖∂tϕ‖L2‖Δ∂tϕ‖L2+C(1+‖u‖H35/6+‖∇ϕ‖L∞)‖∇∂tϕ‖L2,
whence(3.39)ddt∫|∂tϕ|2dx+∫|Δ∂tϕ|2dx≤C‖∂tϕ‖L22+C+C‖u‖H32+C‖∇ϕ‖L∞2.
Here, we have used(3.40)‖∇∂tϕ‖L22≤C‖∂tϕ‖L2‖Δ∂tϕ‖L2.Combining (3.17), (3.21), (3.26), (3.32), (3.34), and (3.39), using Lemmas 2.1, 2.3, and 2.5, we conclude that(3.41)‖u‖L∞(0,T;H3)≤C,(3.42)‖θ‖L∞(0,T;H1)+‖∂tθ‖L∞(0,T;L2)+‖∂tθ‖L2(0,T;H1)≤C,(3.43)‖∂tϕ‖L∞(0,T;L2)≤C.
It follows from (1.3), (3.41), and (3.42) that(3.44)θ∈L∞(0,T;H2)∩L2(0,T;H3).
It follows from (1.4), (1.5), (3.41), and (3.43) that(3.45)‖ϕ‖L∞(0,T;H4)≤C.This completes the proof.



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*Source: 802876-2012-04-09.xml*