# Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces

Authors: Min Fu; Chao Cai
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

This study is focused on the pressure blow-up criterion for a smooth solution of three-dimensional zero-diffusion Boussinesq equations. With the aid of Littlewood-Paley decomposition together with the energy methods, it is proved that if the pressure satisfies the following condition on margin Besov spaces,π(x,t)∈L2/(2+r)(0,T;B˙∞,∞r) for r=±1, then the smooth solution can be continually extended to the interval (0,T⁎) for some T⁎>T. The findings extend largely the previous results.


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## Body

## 1. Introduction and Main Results

It is well known that mathematical models in fluid dynamics have attracted more and more attention in the past ten years [1]. In this paper, we consider the dynamical models of the ocean or the atmosphere which arise from the density dependent incompressible Navier-Stokes equations by using the so-called Boussinesq approximation [2]. The so-called Boussinesq system is governed by the following nonlinear partial differential equations: (1)∂tu-μΔu+u·∇u+∇π=θe3,∂tθ-κΔθ+u·∇θ=0,∇·u=0,associated with the initial conditions (2)ux,0=u0,θx,0=θ0.Here, u(x,t),  θ(x,t), and π(x,t) represent the unknown velocity vector field, temperature field, and the unknown pressure scalar field, respectively. The constant μ>0 is the kinematic viscosity and the constant κ≥0 is the thermal diffusivity. The quantities u0(x) and θ0(x) are the given initial velocity and initial temperature, respectively.As an important mathematical model in many geophysical applications, the Boussinesq system attracted more and more attention in the past ten years [3–6]. Moreover, when the temperature field θ=0, it reduces to the classic Navier-Stokes equations (see [7]) (3)∂tu-μΔu+u·∇u+∇π=0,∇·u=0.In the three-dimensional case, the same as the classic Navier-Stokes equations, the issue on the global smooth solution with large initial data is still a challenging open problem. The development of blow-up criteria is of importance for both theoretical and practical situations [8–10]. When ν,κ>0, the first blow-up criterion of 3D Boussinesq equations in Lebesgue space was considered by Ishimura and Morimoto [11]; they proved that if the velocity satisfies (4)∇u∈L10,T;L∞R3,then the smooth solution can be continually extended to the interval (0,T∗) for some T∗>T. For the zero-viscosity case, that is, μ=0, κ>0, Fan and Zhou [12] studied the blow-up criterion: (5)∇×u∈L10,T;B˙∞,∞0R3.In the zero-diffusion case, that is, μ>0, κ=0, the situation becomes more difficult. The main obstacle is that the temperature function θ(x,t) in the transport equation does not gain any smoothness. Hence, the blow-up issue of the zero-diffusive Boussinesq equations (1) with κ=0 is more difficult than that of full viscous Boussinesq system (1). Jia et al. [13] recently studied the blow-up criterion for local smooth solutions of zero-diffusive Boussinesq equations (1) in the large critical Besov space (6)∫0TuBq,∞sR3pds<∞with (7)2p+3q=1+s,31+s<p≤∞,-1<s≤1,p,s≠∞,1.Recently, Wang [14] also proved the blow-up criterion for the zero-diffusive Boussinesq equation when the velocity components satisfy (8)∫0Tu1+u2Lp,∞q1+ln⁡e+∇uL22ds<∞,2q+3p=1,3<p<∞.For the pressure blow-up criterion of the zero-diffusive Boussinesq equations (1), Dong et al. [15] investigated the pressure regular criterion when (9)πx,t∈L2/2+r0,T;B˙∞,∞rR3for-1<r<1. However, the methods in [15] are not available for the margin case r=1 or r=-1. One may also refer to some important regularity criteria on the fluid dynamics [16, 17].The aim of the present paper is to improve the pressure blow-up criterion for smooth solution of three-dimensional zero-diffusion Boussinesq equations in the margin Besov spacesr=±1 in (9); more precisely, we will prove the following result.Theorem 1.
Supposeμ>0, κ=0. Let T>0; (u,θ) is a smooth solution of zero-diffusion Boussinesq equations (1) with (u0,θ0)∈Hm(R3), m>3/2. If the pressure π satisfies (10)πx,t∈L2/2+r0,T;B˙∞,∞rR3forr=±1,then the smooth solution can be continually extended to the interval (0,T∗) for some T∗>T.Theorem1 also implies the following corollary.Corollary 2.
Suppose(u,θ) is the smooth solution of zero-diffusion Boussinesq equations satisfying (11)u∈C0,T1;HmR3∩L20,T1;Hm+1R3,θ∈C0,T1;HmR3.If T is the maximal existence time of the smooth solution, then (12)T<∞⟹∫0TπB˙∞,∞r2/2+rds=+∞,where r=±1.Remark 3.
It should be mentioned that since our work spaces here are margin cases in Besov spaces, the methods used by Dong et al. [15] where the finding is mainly based on the function spaces decomposition cannot be available any more. Furthermore, compared with the many previous results on the pressure regularity criterion for full viscous fluid dynamical models such as Navier-Stokes equations and MHD equations (see [18]), the zero-diffusion Boussinesq equations (1) do not have the important inequality (13)πLp≤CuL2p2,1<p<∞,due to the appearance of θe3. In order to overcome the difficulty of the absence of the above estimate, we will give some more explicit estimates on the gradient of the pressure in this paper. Additionally, our results here more or less extended the previous results by Gala et al. [19, 20] and others [14]. Since it is also interesting and important to consider this issue in some working space such as Morrey Space (see [17, 21, 22]), we will focus on this problem in the forthcoming paper.


## 2. Preliminary

In this section,C stands for a generic positive constant which may vary from line to line. Lp(R3) with 1≤p≤∞ denotes the usual Lebesgue space and Hs(R3) with s∈R is the inhomogeneous fractional Sobolev space with the norm (14)fHs=∫R3I-Δs/2f2dξ1/2.We now provide the Littlewood-Paley decomposition (see [23]). Let S(R3) be the Schwartz class of a rapidly decreasing function. Given f∈S(R3), we define the Fourier transformation Ff=f^ as (15)f^ξ=∫R3e-ix·ξfxdx.Take two nonnegative radial functionsχ,ψ∈S(R3) supported, respectively, in B={ξ∈R3:|ξ|≤4/3} and C={ξ∈R3:3/4≤|ξ|≤8/3}, such that (16)∑j∈Zψ2-jξ=1,ξ∈R3∖0.Let h=F-1ψ and h~=F-1χ. The frequency localization operator is defined by (17)Δjf=ψ2-jDf=23j∫R3h2jyfx-ydy,j≥0,Sjf=χ2-jDf=∑-1≤k≤j-1Δkf=23j∫R3h~2jyfx-ydy,Δ1f=S0f,Δjf=0forj≤-2.Formally,Δj is a frequency projection to the annulus ξ≈2j, and Sj is a frequency projection to the ball ξ≲2j. The aforementioned dyadic decomposition has nice quasi-orthogonality, with the choice of χ and ψ; namely, given any χ,ψ∈S(R3), we have the following properties: (18)ΔiΔjf=0,ifi-j≥2,ΔiSj-1Δjg=0,ifi-j≥5.With the full-dyadic decomposition, we define the homogeneous Besov spaceB˙p,qs(R3): (19)B˙p,qsR3=f∈S′R3PR3:fB˙p,qs<∞,where (20)fB˙p,qs=∑j=-∞∞2jsqΔjfLpq1/q,1≤q<∞,supj∈Z2jsΔjfLp,q=∞.The set S′(R3) of tempered distributions is the dual set and P(R3) is the polynomials space.In Lemma4, we recall the Bernstein inequalities which will be applied frequently.Lemma 4 (Chemin [24]).
Supposek,j∈Z and 1≤p≤q≤∞; one has, for all f∈S(R3),(21)supα=k∇αΔjfLq≤C2jk+3j1/p-1/qΔjfLp,where C are the positive constants independent of j,  k.Lemma 5 (Meyer [25]).
For any functionf belonging to the homogeneous Sobolev space H˙1(R3), one has (22)fL42≤CfB∇fL2,where B is the homogeneous Besov space B˙∞,∞-1(R3).Lemma 6 (Kato-Ponce [26]).
Supposef,g∈Wk,p(R3), 1<p<∞, for 1≤s≤k; then, (23)Jsfg-fJsgLp≤C∇fLqJs-1gLr+CgLqJsfLr,where Js=(I-Δ)s/2, 1/p=1/q+1/r, and 1<q<∞,  1<r<∞.


## 3. Proof of Theorem1

Step 1 (energy estimate).
Multiplying both sides of the second equation of zero-diffusion Boussinesq equations (1) by |θ|p-2θ and integrating in R3 give (24)1pddtθtLpp=0.Integrating in time, (25)θtLp≤θ0Lp,where we have used ∫R3(u·∇)θθdx=0.
Taking the inner product to the first equation of (1) by u and applying the integration by parts yield (26)12ddtutL22+μ∇uL22≤θL2uL2≤θ0L2uL2;therefore, it follows from Gronwall inequality that (27)sup0<t<TutL22+μ∫0T∇uL22dt≤C.
Since the pressure here plays an important role in our argument, we apply the operator∇div to the first equation of (1) to get (28)∇π=-Δ-1∑i,j=13∂2∂xi∂xj∇uiuj--Δ-1∇∂3θ3,and then, according to Calderon-Zygmund inequality and (27), we obtain (29)∇πLp≤Cu∇uLp+CθLp≤Cu∇uLp+C,1<p<∞.Step 2 (L4 estimate of u).
Taking the inner product to the first equation, the second equation of (1) by |u|2u and |θ|2θ, respectively, employing Hölder inequality and Young inequality, we have (30)14ddtuL44+θL44+μu∇uL22≤∫R3θe3·uu2dx-∫R3u·∇πu2dx≤CθL4uL43-∫R3u·∇πu2dx≤CuL44+θL44-∫R3u·∇πu2dx=CuL44+θL44+I,where we have used (31)∫R3u·∇u·u2udx=0,∫R3u·∇θ·θ2θdx=0.Case 1 (r=1).
That is,(32)∫0TπtB˙∞,∞12/3dt<∞,I≤∫R3u·∇πu2dx=∫R3∑j<-N+∑j=-NN+∑j>NΔj∇π·u2udx≤∫R3∑j<-NΔj∇π·u2udx+∫R3∑j=-NNΔj∇π·u2udx+∫R3∑j>NΔj∇π·u2udx=I1+I2+I3for an arbitrary positive integer N.In order to estimate the termI1, we apply Hölder inequality, Lemma 5, and Young inequality: (33)I1≤∑j<-NΔj∇πL4uL43≤∑j<-N23j1/2-1/4Δj∇πL2uL43≤∑j<-N23/2j1/2∑j<-NΔj∇πL221/2uL43≤2-3N/4∇πL2uL43≤2-3N/4u∇uL2+CuL43≤C2-N/4uL43u∇uL2+C2-N/4uL43.In order to estimate the termI2, we apply Hölder inequality and Young inequality: (34)I2≤∑j=-NNΔj∇πL4uL43≤N3/4∇πB∞,∞01/2∇πL21/2uL43≤CN∇πB∞,∞02/33/4u∇uL21/2+CuL43≤CN∇πB∞,∞02/3uL44+μ8u∇uL22+C.ForI3, we apply Hölder inequality, Lemma 5, and (29): (35)I3≤∑j>NΔj∇πL4uL43≤2-3N/4∇πL2uL43≤C2-N/4uL43u∇uL2+C≤C2-N/4uL43u∇uL2+C2-N/4uL43.We only need to choose the integer N satisfying (36)C2-N/4uL43≤μ8.For example, we may set (37)N=1/3ln⁡μ/8C+ln⁡uL4+eln⁡2+1.CombiningI1,  I2, and  I3 and the above inequality yields (38)ddtuL44+θL44+μu∇uL22≤CuL44+θL44+CN∇πB∞,∞02/3uL44+C,and the above inequality reduces to (39)ddtuL44+θL44≤CuL44+θL44+CN∇πB∞,∞02/3uL44+C=C+ClnC+lnuL44+eln⁡2+1∇πB∞,∞02/3uL44+θL44+C.Taking Gronwall inequality into consideration, (40)uL44+θL44≤u0L44+θ0L44+CTexp⁡CT+∫0TClnC+ln⁡uL44+eln2+1∇πB∞,∞02/3dτ≤u0L44+θ0L44+CTexp⁡CT+∫0TCln⁡uL44+e+C∇πB∞,∞02/3dτ,for a suitable constant C.Thus, we also have(41)ln⁡uL44+θL44+e≤ln⁡u0L44+θ0L44+CT+CT+∫0TCln⁡uL44+e+C∇πB∞,∞02/3dτ.Employing Gronwall inequality again, one shows that (42)ln⁡uL44+θL44+e≤ln⁡u0L44+θoL44+CT+Cexp⁡∫0TC∇πB∞,∞02/3dτ<∞.Case 2 (r=-1).
That is,(43)∫0TπtB˙∞,∞-12dt<∞.Employing Hölder inequality, Young inequality, and Lemma5, one shows that (44)I=2∫R3πuu·∇udx≤CπL42uL42+μ8u∇uL22≤CπB∞,∞-1∇pL2uL42+μ8u∇uL22≤CπB∞,∞-1C+u∇uL2uL42+μ8u∇uL22≤CπB∞,∞-12uL44+μ4u∇uL22+C.Inserting it into (30), we get (45)ddtuL44+θL44+μu∇uL22≤CθL44+uL44+CπB∞,∞-12uL44≤C+CπB∞,∞-12uL44+θL44.Taking Gronwall inequality into consideration, one shows that(46)sup0<t<TutL44+θL44+μ∫0Tu∇uL22dt≤u0L44+θ0L44exp∫0TC+CπtB˙∞,∞-12dt≤Cu0L44+θ0L44exp⁡∫0TπtB˙∞,∞-12dt<∞.From the estimates ofI in both Cases 1 and 2, we now derive the L4 bound of the velocity u: (47)uL∞0,T1;L4≤C.Step 3 (H1 estimates of u).
Taking the inner product to the first equation of (1) by Δu and applying the integral by parts yield(48)12ddt∇uL22+μΔuL22=∫R3u·∇u·Δudx-∫R3θe3·Δudx≕J1+J2.For J1, J2, we employ Hölder inequality, Young inequality, and Gagliardo-Nirenberg inequality: (49)J1=∫R3u·∇u·Δudx≤CuL4∇uL4ΔuL2≤CuL4∇uL21/4ΔuL27/4≤μ4ΔuL22+C∇uL22,(50)J2≤CθL2ΔuL2≤μ4ΔuL22+CθL22≤μ4ΔuL22+C.Plugging (49) and (50) into (48) to deduce (51)ddt∇uL22+μΔuL22≤C∇uL22+C,taking Gronwall inequality into consideration together with (27), it follows that (52)sup0<t<T∇utL22+∫0TΔuL22≤C;that is (53)sup0<t<TutH12+∫0T∇uH12≤C.Step 4 (Hm estimates of u, θ).
To prove theHm estimates of u, θ, we need L∞ estimates of ∇u, ∇θ. To do so, we first rewrite the first equations of (1): (54)∂tu-μΔu=θe3-u·∇u-∇π.Thanks to (55)∫0Tθe3L32ds≤θ0L32T,∫0Tu·∇uL32ds≤∫0TuL62∇uL22ds≤esssup0<s<T∇usL22∫0TΔuL22ds≤C,∫0T∇pL32ds≤∫0Tu·∇uL32ds+CT≤esssup0<s<T∇usL22∫0TΔuL22ds+CT≤CT,the maximal regularity properties of the heat equation allow us to derive (56)∫0TuW2,32ds≤Cimplied by Sobolev imbedding inequality (57)∫0T∇uL∞ds≤C.
Now, we take the operator∇ into the second equation of (1) and then take the inner product by |∇θ|p-2∇θ to give (58)ddt∇θLp≤C∇uL∞∇θLp.Employing Gronwall inequality yields (59)∇θLp≤∇θ0Lpexp⁡∫0T∇uL∞dτ.
For any givenϵ,ρ>0, we define the set Aϵ,ρ(t):(60)Aϵ,ρt=x∈R3;∇θ>∇θL∞-ε,x>ρ,and then (61)∇θtL∞-ϵAϵ,ρt≤∇θLp≤∇θ0L22/p∇θ0L∞1-2/pexp⁡∫0T∇uL∞dτ≤C,where Aϵ,ρt is the Lebesgue measure of the set Aϵ,ρ(t). Since C is independent of t,  ϵ, the above uniform bound allows us to take the limitation t→∞, ϵ→0; then, we have (62)∇θtL∞≤∇θ0L∞exp⁡∫0T∇uL∞dτ.
Taking the derivative operatorJm=(I-Δ)m/2 into the first equation of (1) and taking the inner product by Jmu and then applying Lemma 6, Hölder inequality, and Young inequality,(63)12ddtuHm2+μ∇uHm2≤∫R3Jmu·∇u-u·∇JmuJmudx+∫R3Jmθe3Jmudx≤C∇uL∞JmuL22+CJmθL2JmuL2≤C∇uL∞uHm2+θHm2+uHm2.
Besides, we employ similar methods to (63) to the second equation of (1), yielding (64)12ddtθHm2≤∫R3Jmu·∇θ-u·∇JmθJmθdx≤C∇uL∞JmθL22+C∇θL∞JmθL2JmuL2≤C∇uL∞+∇θL∞θHm2+C∇θL∞uHm2.Combining (63) and (64), (65)ddtuHm2+θHm2+2μ∇uHm2≤C∇uL∞+∇θL∞+CuHm2+θHm2,and taking Gronwall inequality into consideration give (66)sup0<t<TutHm2+θtHm2+2μ∫0T∇uHm2dτ≤C.
Thus, we complete the proof of Theorem1.



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