Regularity Criterion for the Three-dimensional Boussinesq Equations
aa r X i v : . [ m a t h . A P ] J un REGULARITY CRITERION FOR THE THREE-DIMENSIONALBOUSSINESQ EQUATIONS
KAREN ZAYA
Abstract.
We prove that a solution ( u, θ ) to the three-dimensional Boussi-nesq equations does not blow-up at time T if k u ≤ Q k B ∞ , ∞ is integrable on(0 , T ), where u ≤ Q represents the low modes of Littlewood-Paley projection ofthe velocity u . Introduction
Consider the three-dimensional incompressible Boussinesq equations, ∂ t u + ( u · ∇ ) u = −∇ p + ν ∆ u + θe , (1.1) ∂ t θ + ( u · ∇ ) θ = κ ∆ θ, (1.2) ∇ · u = 0 , (1.3)with initial data u ( x,
0) = u ( x ) , (1.4) θ ( x,
0) = θ ( x ) , (1.5)where x ∈ R , t ≥ u = u ( x, t ) is the velocity vector with divergence-free initialdata, p = p ( x, t ) is the pressure scalar, and θ = θ ( x, t ) is the temperature scalar.The fluid kinematic viscosity is ν ≥
0, the thermal diffusivity is κ ≥
0, and e =(0 , , T . When θ vanishes, the system reduces to the incompressible Navier-Stokesequations, which can be further reduced to the incompressible Euler equations bysetting ν = 0.The Boussinesq equations are important in the study of atmospheric sciencesand they yield a wealth of interesting and difficult problems from a mathematicalperspective. The regularity of (1.1)-(1.3) has been studied throughly on its own andin relation to the regularity of other equations, such as the Navier-Stokes equations,Euler equations, and magneto-hydrodynamics (MHD) equations.Regularity criteria for the Boussinesq and related equations mentioned above canbe split into different classes, one of which is Beale-Kato-Majda-like (abbreviated“BKM-like”) criteria. The original result by Beale, Kato, and Majda [2] placed acondition on the vorticity in the Navier-Stokes equations. They proved if Z T k∇ × u k L ∞ d t < ∞ , (1.6) Department of Mathematics. University of Michigan. 2074 East Hall. 530 ChurchStreet. Ann Arbor, MI 48109-1043
E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases.
Boussinesq equations, regularity, dissipation wave number. then a smooth solution to the Navier-Stokes equations on (0 , T ) does not blow upat time T . This condition was weakened for the Euler equations by Planchon [16]and improved for the three-dimensional Navier-Stokes equations by Cheskidov andShvydkoy [9]. In [7], Cheskidov and Dai developed BKM-like, but weaker, regularitycriterion for the three-dimensional MHD equations.A related family of regularity criteria fall into the Ladyzhenskaya-Prodi-Serrincategory. For the Navier-Stokes equations, the condition is u ∈ L p (0 , T ; L r ) , for 2 p + 3 r = 1 , r ∈ (3 , ∞ ] . (1.7)Various extensions and improvements of this type of criteria have been developedsince then, such as the extension to the case for r = 3 by Escauriaza, Seregin, andˇSver´ak [15] and extensions to Besov spaces.Both kinds of regularity criteria were developed for the three-dimensional incom-pressible Boussinesq equations. In [18] and [19], Qiu, Du, and Yao developed Serrin-type blow-up criteria for the Boussinesq equations. Particularly in [18], the authorsshowed a smooth solution to (1.1)-(1.3) on time interval [0 , T ) will remain smoothat time T if u ∈ L q (cid:0) , T ; B sp, ∞ ( R ) (cid:1) for q + p = 1 + s , s +1 < p ≤ ∞ , − < s ≤ p, s ) = ( ∞ , ∇ u ∈ L (cid:0) , T ; L ∞ ( R ) (cid:1) . Later Fan and Zhou [10] studied theBoussinesq equations with partial viscosity and proved BKM-like regularity criteriain terms of the vorticity: ∇ × u ∈ L (cid:0) , T ; ˙ B ∞ , ∞ ( R ) (cid:1) . More regularity conditionsin the three-dimensional case were developed in the following years, some of whichcan be found in [17], [21], [22], [24], [25], and [26].A great deal of literature has also been produced for the two-dimensional case.We will not discuss these results, but rather refer the reader a few publications onthat topic: [1], [3], [4], [5], [11], [12], [13], [20], and [23].The main theorem discussed here also falls into the two main kinds of regularitycriteria discussed above. We will show: Theorem 1.1.
Let (cid:0) u, θ (cid:1) be a weak solution to (1.1) - (1.3) on [0 , T ] , assume (cid:0) u, θ (cid:1) is regular on (0 , T ) , and k u ≤ Q k B ∞ , ∞ ∈ L (0 , T ) . (1.8) Then (cid:0) u ( t ) , θ ( t ) (cid:1) is regular on (0 , T ] .Remark . We note that the above BKM-like regularity criterion also recoversthe previous known Prodi-Serrin-type regularity, in particular we improve upon theresults in [18], by recovering the whole range, including the endpoint ( p, s ) = ( ∞ , u . Remark . The notation u ≤ Q denotes the low modes of u and B ∞ , ∞ is a Besovspace. More precise definitions are presented in the next section (see (2.3) andDefinition 2.2). 2. Background
Littlewood-Paley Decomposition.
We utilize Littlewood-Paley decompo-sition in our proof. Denote wave numbers as λ q = 2 q (in some wave units). For EGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS 3 ψ ∈ C ∞ ( R ), define ψ ( ξ ) = (cid:26) , for | ξ | ≤ , for | ξ | ≥ . Next define φ ( ξ ) = ψ ( ξ/ λ ) − ψ ( ξ ) and φ q ( ξ ) = (cid:26) φ ( ξ/ λ q ) , for q ≥ ψ ( ξ ) , for q = − . The Littlewood-Paley projection operator ∆ q is defined as∆ q u = Z R u ( x − y ) F − ( φ q )( y ) d y , (2.1)where F − is the inverse Fourier transform. Primarily, we will denote ∆ q u , the q th Littlewood-Paley piece of u , as u q instead. In the sense of distributions, one has u = ∞ X q = − u q . (2.2)We also define u ≤ Q = Q X q = − u q , u ≥ Q = ∞ X q = Q u q , (2.3)and ˜ u q = u q − + u q + u q +1 , which will be useful notation for the proof.2.2. Notation, Spaces, and Solutions.
We will use the symbol . (or & ) tomean that an inequality holds up to an absolute constant, we will denote L p -normsas k · k p , and ( · , · ) will denote the L inner product.We use Littlewood-Paley decomposition to define some useful norms. Regularity(see Definition 2.4 ) is defined via Sobolev norms. Definition 2.1.
The homogeneous Sobolev space ˙ H s contains functions u suchthat the associated norm k u k ˙ H s = (cid:16) ∞ X q = − λ sq k u q k (cid:17) , (2.4)for s ∈ R , is finite. Note that k u k H s ∼ k u k ˙ H s , which we will keep in mind through-out the work below.The regularity criterion is defined in terms of the following Besov norm: Definition 2.2.
The norm of u in Besov space B ∞ , ∞ is defined as k u ( t ) k B ∞ , ∞ = sup q ≥− λ q k u q ( t ) k ∞ . (2.5)The Besov space B ∞ , ∞ is the space of tempered distributions u such that k u ( t ) k B ∞ , ∞ is finite.We work in the class of weak solutions: REGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS
Definition 2.3.
A weak solution of (1.1)-(1.3) on [0 , T ] is a pair of functions ( u, θ ), u divergence free, in the class u, θ ∈ C w (cid:0) [0 , T ]; L ( R ) (cid:1) ∩ L (cid:0) , T ; H ( R ) (cid:1) such that (cid:0) u ( t ) , φ ( t ) (cid:1) − (cid:0) u , φ (0) (cid:1) = Z t (cid:0) u ( s ) , ∂ s φ ( s ) (cid:1) + ν (cid:0) u ( s ) , ∆ φ ( s ) (cid:1) + (cid:0) u ( s ) · ∇ φ ( s ) , u ( s ) (cid:1) + (cid:0) θ ( s ) e , φ ( s ) (cid:1) d s and (cid:0) θ ( t ) , φ ( t ) (cid:1) − (cid:0) θ , φ (0) (cid:1) = Z t (cid:0) θ ( s ) , ∂ s φ ( s ) (cid:1) + κ (cid:0) θ ( s ) , ∆ φ ( s ) (cid:1) + (cid:0) u ( s ) · ∇ φ ( s ) , θ ( s ) (cid:1) d s , for all divergence free test functions φ ∈ C ∞ (cid:0) [0 , T ] × R (cid:1) . Definition 2.4.
A Leray-Hopf weak solution of (1.1)-(1.5) is regular on time in-terval I if the Sobolev norm k u k H s is continuous for s ≥ on I . Remark . One can apply a standard bootstrap argument to show if a solutionis regular, then u and θ are smooth.2.3. The Dissipation Wave Number.
The development of our regularity cri-terion is linked to Kolmogorov’s theory of turbulence and the dissipation wavenumber. The dissipation wave number is a time-dependent function that separatesthe low frequency inertial range, where the nonlinear term dominates the dynam-ics, from the high frequency dissipative range, where viscous forces takeover. In [9],Cheskidov and Shvydkoy defined the dissipation wave number and proved that if itbelongs to L / (0 , T ), then the solution of the Navier-Stokes equations is regular upto time T . They also showed that the dissipation wave number belongs to L (0 , T )for every Leray-Hopf solution.We define the dissipation wave number Λ( t ) for the Boussinesq equations in ananalogous manner: Q ( t ) = min { q : λ − p k u p k ∞ < c min { ν, κ } , ∀ p > q, q ≥ } , (2.6) Λ( t ) = λ Q ( t ) , (2.7)for absolute constant c > Proof of the Main Theorem
We prove Theorem 1.1 in this section.
Proof.
We test (1.1) with ∆ q u and (1.2) with ∆ q θ , which yields12 dd t k u q k ≤ − ν k∇ u q k + Z R ∆ q (cid:0) u · ∇ u (cid:1) · u q d x − Z R ∆ q (cid:0) θe (cid:1) · u q d x , (3.1) 12 dd t k θ q k ≤ − κ k∇ θ q k + Z R ∆ q (cid:0) u · ∇ θ (cid:1) · θ q d x , (3.2) EGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS 5 and then multiply (3.1) by λ sq and (3.2) by λ σq , add those two equations together,and sum over q to arrive at(3.3) 12 dd t ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) ≤ − ∞ X q = − (cid:0) λ sq ν k∇ u q k + λ σq κ k∇ θ q k (cid:1) + I + I + I , where I = ∞ X q = − λ sq Z R ∆ q (cid:0) u · ∇ u (cid:1) · u q d x , (3.4) I = − ∞ X q = − λ sq Z R ∆ q (cid:0) θe (cid:1) · u q d x , (3.5) I = ∞ X q = − λ σq Z R ∆ q (cid:0) u · ∇ θ (cid:1) · θ q d x . (3.6)For (3.6), we use a similar method as in [7]. First, we decompose (3.6) into threeparts: I = ∞ X q = − X | q − p |≤ λ σq Z R ∆ q ( u ≤ p − · ∇ θ p ) θ q d x + ∞ X q = − X | q − p |≤ λ σq Z R ∆ q ( u p · ∇ θ ≤ p − ) θ q d x (3.7) + ∞ X q = − X p ≥ q − λ σq Z R ∆ q ( u p · ∇ ˜ θ p ) θ q d x = I , + I , + I , . We use H¨older’s inequality on I , to find | I , | . ∞ X q = − λ σq k u q k ∞ X | q − p |≤ k θ p k X p ′ ≤ p − λ p ′ k θ p ′ k . Then we split the sum into high and low modes. For the high modes we use thedefinition of Λ( t ) and for the low modes we use f ( t ) = k u ≤ Q ( t ) ( t ) k B ∞ , ∞ = sup q ≤ Q ( t ) λ q k u q ( t ) k ∞ , (3.8) REGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS (which will be used to define the regularity criterion later) to find | I , | . cκ ∞ X q = Q +1 λ σ +1 q X | q − p |≤ k θ p k X p ′ ≤ p − λ p ′ k θ p ′ k + f ( t ) Q X q = − λ σ − q X | q − p |≤ k θ p k X p ′ ≤ p − λ p ′ k θ p ′ k . cκ ∞ X q = Q − λ σ +1 q k θ q k X p ′ ≤ q λ p ′ k θ p ′ k + f ( t ) Q +2 X q = − λ σ − q k θ q k X p ′ ≤ q λ p ′ k θ p ′ k . After a rearrangement and an application of Jensen’s inequality, we arrive at | I , | . cκ ∞ X q = Q − λ σ +1 q k θ q k X p ′ ≤ q λ σq − p ′ λ σ +1 p ′ k θ p ′ k + f ( t ) Q +2 X q = − λ σq k θ q k X p ′ ≤ q λ σ − q − p ′ λ σp ′ k θ p ′ k . cκ ∞ X q = − λ σ +2 q k θ q k + f ( t ) Q +2 X q = − λ σq k θ q k , for σ < . (3.9)By Bony’s paraproduct and commutator notation, which says[∆ q , u ≤ p − · ∇ ] θ p = ∆ q ( u ≤ p − · ∇ θ p ) − u ≤ p − · ∇ ∆ q θ p , one may decompose I , as I , = ∞ X q = − X | q − p |≤ λ σq Z R [∆ q , u ≤ p − · ∇ ] θ p θ q d x + ∞ X q = − λ σq Z R u ≤ q − · ∇ θ q θ q d x (3.10) + ∞ X q = − X | q − p |≤ λ σq Z R ( u ≤ p − − u ≤ q − ) · ∇ ∆ q θ p θ q d x = I , , + I , , + I , , . In [7], they note that their term (equivalent to our I , , ) can be estimated as | I , , | . cκ ∞ X q = Q +2 λ σ +2 q k θ q k + f ( t ) ∞ X q = − λ σq k θ q k . EGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS 7
The second term, I , , = 0 because of the divergence-free condition on u . We alsorefer the reader to [7], where one can find | I , , | + | I , | . cκ ∞ X q = − λ σ +2 q k θ q k + f ( t ) Q +2 X q = − λ σq k θ q k . The above estimates on the pieces of (3.6) yield | I | . cκ ∞ X q = − λ σ +2 q k θ q k + f ( t ) ∞ X q = − λ σq k θ q k . (3.11)For (3.4), we refer the reader to the estimates carried out in [9] on the Navier-Stokes equations (and the equivalent term in [7] on the MHD equations), wherethey show | I | . cν ∞ X q = − λ s +2 q k u q k + f ( t ) ∞ X q = − λ sq k u q k , (3.12)where f ( t ) is the same as in (3.8). The bound (3.12) holds for12 ≤ s < . (3.13)We use Young’s inequality to estimate (3.5) as | I | = (cid:12)(cid:12)(cid:12) ∞ X q = − λ sq Z R ∆ q (cid:0) θe (cid:1) · u q d x (cid:12)(cid:12)(cid:12) . ∞ X q = − λ sq (cid:0) k u q k + k θ q k (cid:1) . (3.14)We may break up this sum as follows: | I | . ∞ X q = − λ sq k u q k + N X q = − λ sq k θ q k + ∞ X q = N λ sq k θ q k , (3.15)where N is finite and will be determined later.Estimates (3.12), (3.15), and (3.11) in (3.3) yield12 dd t ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) . − ν ∞ X q = − λ s +2 q k u q k + cν ∞ X q = − λ s +2 q k u q k − κ ∞ X q = − λ σ +2 q k θ q k + cκ ∞ X q = − λ σ +2 q k θ q k + N X q = − λ sq k θ q k + ∞ X q = N λ sq k θ q k (3.16) + (cid:0) f ( t ) + 1 (cid:1) ∞ X q = − λ sq k u q k + f ( t ) ∞ X q = − λ σq k θ q k . For 2 s < σ + 2(3.17) REGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS and absolute constants C , C , C , C and C , we have12 dd t ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) ≤ − ν ∞ X q = − λ s +2 q k u q k + C cν ∞ X q = − λ s +2 q k u q k − κ ∞ X q = − λ σ +2 q k θ q k + C cκ ∞ X q = − λ σ +2 q k θ q k + C N X q = − λ sq k θ q k + C λ s − σ − N ∞ X q = N λ σ +2 q k θ q k (3.18) + C (cid:0) f ( t ) + 1 (cid:1) ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) . For N large enough, one may choose N = N ( κ ) such that C λ s − σ − N ≤ κ/
2. Infact, one may solve for this N explicitly. Since such a finite N exists, then we canuse the following two facts: N X q = − λ sq k θ q k < ∞ , (3.19)and C λ s − σ − N ∞ X q = N λ σ +2 q k θ q k ≤ κ ∞ X q = − λ σ +2 q k θ q k . (3.20)Using (3.19) and (3.20) in (3.18), we arrive at the following differential inequality:12 dd t ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) ≤ ( − ν + C cν ) ∞ X q = − λ s +2 q k u q k ( − κ C cκ ) ∞ X q = − λ σ +2 q k θ q k + C (cid:0) f ( t ) + 1 (cid:1) ∞ X q = − (cid:0) λ sq k u q k + λ σq k θ q k (cid:1) (3.21) + C N X q = − λ sq k θ q k The choice c = min { C , C } yields(3.22) dd t (cid:0) k u k H s + k θ k H σ (cid:1) ≤ C ( ν, κ, s, σ ) (cid:0) f ( t ) + 1 (cid:1)(cid:0) k u k H s + k θ k H σ (cid:1) + C N X q = − λ sq k θ q k . By Gr¨onwall’s inequality (noting (3.19)), we have that k u k H s + k θ k H σ remainsbounded on (0 , T ) for ≤ s < s − < σ < k u ≤ Q k B ∞ , ∞ ∈ L (0 , T ) . (3.23)Thus, by Definition 2.4, we reach the desired conclusion. (cid:3) EGULARITY CRITERION FOR 3D BOUSSINESQ EQUATIONS 9
Acknowledgments:
The work of K. Zaya was supported by NSF grants DMS-1210896 and DMS-1517583 while at the University of Illinois at Chicago, and laterby NSF grant DMS-1515161 while at the University of Michigan.
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