The Boussinesq Equations
HomeSearchCollectionsJournalsAboutContact usMy IOPscienceWell-posedness and inviscid limits of the Boussinesq equations with fractional LaplaciandissipationThis content has been downloaded from IOPscience. Please scroll down to see the full text.2014 Nonlinearity 27 )View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 139.78.143.171This content was downloaded on 07/08/2014 at 01:40Please note that terms and conditions apply.
London Mathematical SocietyNonlinearity 27 (2014) 2215Well-posedness and inviscid limits of theBoussinesq equations with fractionalLaplacian dissipationJiahong Wu1 and Xiaojing Xu21 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences,Stillwater, OK 74078, USA2 School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematicsand Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of ChinaE-mail: jiahong@math.okstate.edu and xjxu@bnu.edu.cnReceived 29 May 2013, revised 1 July 2014Accepted for publication 16 July 2014Published 6 August 2014Recommended by K OhkitaniAbstractThis paper is concerned with the global well-posedness and inviscid limits ofseveral systems of Boussinesq equations with fractional dissipation. Threemain results are proven. The first result assesses the global regularity of twosystems of equations close to the critical 2D Boussinesq equations. This isachieved by examining their inviscid limits. The second result relates the globalregularity of a general system of d-dimensional Boussinesq equations to that ofits formal inviscid limit. The third obtains the global existence, uniqueness andinviscid limit of a system of 2D Boussinesq equations with the Yudovich-typeinitial data.Keywords: Boussinesq equations, fractional Laplacian, global regularity,inviscid limitsMathematics Subject Classification: 35Q35, 35B35, 35B65, 76D031. IntroductionThis paper studies the global regularity and inviscid limits of several Boussinesq systemsof equations with dissipation given by a fractional Laplacian. The Boussinesq equationsconcerned here model large-scale atmospheric and oceanic flows and also play important rolesin the study of Rayleigh–Bénard convection (see, e.g., [14, 19, 31, 37]). Our goal here isseveral fold: first, to establish the global regularity of two systems of Boussinesq equationsthat are close to the 2D Boussinesq equations with critical dissipation through the study of0951-7715/14/092215 18 33.00 2014 IOP Publishing Ltd & London Mathematical SocietyPrinted in the UK2215
Nonlinearity 27 (2014) 2215J Wu and X Xutheir inviscid limits; second, to prove a connection between the global regularity of a generalsystem of d-dimensional Boussinesq equations and that of its formal inviscid limit; and third,to obtain the global existence and uniqueness as well as the inviscid limit of a system of 2DBoussinesq equations with the Yudovich-type initial data.Our first result was mainly motivated by a recent progress on the global regularityissue concerning the 2D Boussinesq equations with fractional Laplacian dissipation orwith partial dissipation. Due to their mathematical significance, these 2D equations haveattracted considerable attention in the last few years (see, e.g., [1–3, 6, 8–10, 15–18, 20–26, 28–30, 33, 35, 36, 42]). Mathematically the 2D Boussinesq equations serve as a lower dimensionalmodel of the 3D hydrodynamics equations. In fact, the Boussinesq equations retain somekey features of the 3D Navier–Stokes and the Euler equations such as the vortex stretchingmechanism. As pointed out in [32], the inviscid Boussinesq equations can be identified withthe 3D Euler equations for axisymmetric flows. One main pursuit has been to establish theglobal regularity of the following 2D Boussinesq system with minimal dissipation α t u u · u ν u p θ e2 , t θ u · θ κ β θ 0,(1.1) · u 0,where u : R2 R2 is a vector field denoting the velocity, θ : R2 R is a scalar functiondenoting the temperature in the content of thermal convection and the density in the modellingof geophysical fluids, e2 is the unit vector in the x2 direction, ν 0 denotes the viscosity,κ 0 denotes the thermal diffusivity, and α [0, 2] and β [0, 2] are real parameters. Herewe adopt the convention that α 0 or β 0 implies the corresponding dissipative term isset to zero. In addition, denotes the Zygmund operator (see [39]), which can bedefined through the Fourier transform, (ξ ) ξ f (ξ ). fThere are geophysical circumstances in which the Boussinesq equations with fractionalLaplacian arise. Flows in the middle atmosphere travelling upwards undergo changes dueto the changes in atmospheric properties, although the incompressibility and Boussinesqapproximations are applicable. The effect of kinematic and thermal diffusion is attenuatedby the thinning of atmosphere. This anomalous attenuation can be modelled using the spacefractional Laplacian (see [7, 19]).Quite a few papers have been devoted to (1.1) and the most recent work targets the criticaland the supercritical cases (see, e.g., [10, 15, 16, 20–26, 28, 33, 42]). In two papers [23, 24]Hmidiet al were able to show the global regularity of (1.1) for two critical cases: (1.1) withν 0, α 1 and κ 0, and (1.1) with ν 0, κ 0 and β 1. Miao and Xue [33] obtainedthe global regularity for (1.1) with ν 0, κ 0 and 6 67 2 6α(1 α)α , 1 , β 1 α, minα 2, , 2 2α.456 2αIn addition, Constantin and Vicol [15] verified the global regularity of (1.1) with2α (0, 2), β (0, 2),β .2 αThe global regularity for the general critical caseα (0, 1),β (0, 1),α β 1appears to be open at this moment. It is worth remarking that success has also been achievedon the global regularity issue beyond the critical case. Hmidi [20] proved the global regularity2216
Nonlinearity 27 (2014) 2215J Wu and X Xufor the 2D Boussinesq equations with logarithmically supercritical dissipation while Chae andWu [10] obtained the global regularity of a generalized Boussinesq equation with the velocitydetermined by the vorticity via an operator logarithmically more singular than the Biot–Savartlaw. Our first result assesses the global regularity of two systems of equations close to thecritical 2D Boussinesq equations. This is achieved by studying the inviscid limits of theseequations and combining with the known global regularity result of the critical Boussinesqequations. The details are given in section 3.The second main result relates the global regularity of a general d-dimensional Boussinesqequations to that of its formal inviscid limit. A special case of this result states that if the 3Dinviscid Boussinesq equations have a classical solution on [0, T ], then any 3D dissipativeBoussinesq equations with viscosity or thermal diffusivity in a suitable range also possess aclassical solution on [0, T ]. This result extends the work of Constantin on the Euler and theNavier–Stokes equations [13]. We defer the precise statement and the proof to section 4.The last part of this paper examines the inviscid limit of the following 2D Boussinesqequations with Yudovich-type initial data, t u u · u ν α u p θ e2 , θ u · θ β θ 0,t(1.2) · u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x),where ν 0, α (0, 1] and β (1, 2]. Here the Yudovich-type initial data refer toθ0 L2 (R2 ) Lq (R2 ) and the initial vorticity ω0 u0 Lq (R2 ) L (R2 ), where2q β 1. Previously Danchin and Paicu [16] studied the global well-posedness of (1.2)with the Yudovich-type data for the case when ν 0 and β 2. In addition, Hmidi andZerguine [25] studied the global regularity of (1.2) with ν 0 and β (1, 2], but with aninitial data (u0 , θ0 ) in a more regular functional setting. As our first step, we establish theglobal existence and uniqueness of solutions to (1.2) with either ν 0 or ν 0. In particular,the solutions are shown to obey global bounds independent of ν in the functional settingω L ([0, T ]; Lq (R2 ) L (R2 )),1 2θ L ([0, T ]; L2 (R2 ) Lq (R2 )) L1 ([0, T ]; Br,1 r (R2 )),where T 0 is arbitrarily fixed and r (2, q). Combining these global bounds with theYudovich approach allows us to show that the difference between a solution (u(ν) , θ (ν) ) of(1.2) with ν 0 and the corresponding solution of (u, θ ) of (1.2) with ν 0 satisfies (u(ν) , θ (ν) )(·, t) (u, θ )(·, t) L2 C(T ) (νt)e b(t)(1.3)for any T 0 and t T , where C(T ) is a constant depending on T and the initial norm only,andtb(t) C ω(·, τ ) L θ(·, τ ) 01 2Br,1 rdτ.Clearly the convergence rate in (1.3) deteriorates as time evolves and may not be improvedwhen the initial data is of the Yudovich type. The precise statement and detailed proof of theseresults are provided in section 5.In addition to the sections containing the main results, section 2 presents some preliminaryfacts and estimates to be used in the subsequent sections. In addition, an appendix on some ofthe functional spaces used in this paper and the Osgood inequality is provided.2217
Nonlinearity 27 (2014) 2215J Wu and X Xu2. Preliminary estimatesThis section contains an upper bound for the solution of an ordinary differential equation(ODE) and a regularity criteria for a general Boussinesq equations. These results will be usedin the subsequent sections.The following lemma gives a global upper bound for the solution of an ODE involving asmall parameter. It slightly extends a previous result of Constantin (see [13, p 315]).Lemma 2.1. Let γ 0 and G 0 be parameters. Let T 0. Let F1 and F2 be nonnegativecontinuous functions on [0, T ]. Consider the ODE dY (t) γ F F Y G Y 2 ,12dt(2.1) Y (0) 0.If we setγ0 8T G1 TF1 (τ )e0 TτF2 (s) ds,dτthen, for any γ (0, γ0 ) and t [0, T ], any solution to (2.1) obeys TT t3e t F2 (τ ) dτF1 (τ )e τ F2 (s) ds dτ ., 12γY (t) min2T G0(2.2)Proof. Obviously (2.1) is equivalent to td t F2 (τ ) dτ (γ F1 GY 2 )e 0 F2 (τ ) dτ .Ye 0dt tOr, in terms of U Y e 0 F2 (τ ) dτ , t tdU γ F1 e 0 F2 (τ ) dτ G e 0 F2 (τ ) dτ U 2 .dtBy lemma 1.3 of [13], if we setγ0 8T Ge T0TF2 (τ ) dτF1 (τ )e τ0 1F2 (s) dsdτ,0then U (t) minTherefore, Y (t) e t03e F2 (τ ) dτ T0TF2 (τ ) dτ2T G, 12γF1 (τ )e τ0F2 (s) dsdτ .0U (t) obeys (2.2). This completes the proof of lemma 2.1. For the convenience of later applications, we state a local existence and regularity criterionfor the following general d-dimensional Boussinesq equations: t u u · u ν α u p θ ed , θ u · θ κ β θ 0,t(2.3) · u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x),where ν 0, κ 0, α 0 and β 0 are real parameters, and ed is the unit vector in thedirection of the last coordinate axis.2218
Nonlinearity 27 (2014) 2215J Wu and X XuLemma 2.2. Let ν 0, κ 0, α 0 and β 0 be real parameters. Assume that(u0 , θ0 ) H s (Rd ) with s 1 d2 . Then there exists T T ( (u0 , θ0 ) H s ) 0 such that (2.3)has a unique solution (u, θ ) on [0, T ] satisfying (u, θ ) C([0, T ]; H s ). In addition, if wefurther know thatT0 u(τ ) L dτ (2.4)0for T0 T , then the solution (u, θ ) can be extended to [0, T0 ].Proof. For the self-containedness, we briefly explain the lines of proof. The local wellposedness of (3.1) can be established through a standard procedure such as the Picard-typetheorem (see, e.g., [32]). To prove the regularity criterion, we obtain by standard energyestimates that d u 2H s θ 2H s C u 2 s α2 C θ 2 s βHH 2dt C(1 u L θ L ) u 2H s θ 2H s ,(2.5)where C’s are constants. In addition, it follows from the θ -equation that, for any t 0,t θ (t) L θ0 L exp u L dτ .0Therefore, (2.4) implies that θ L ([0, T0 ]; L (Rd )). Gronwall’s inequality appliedto (2.5) then leads to a bound for (u, θ )(·, T0 ) H s and thus the desired extension. Thiscompletes the proof of lemma 2.2. 3. 2D Boussinesq equations close to the critical equationsThis section studies the global well-posedness and the inviscid limits of two systems ofequations close to the critical Boussinesq equations. First we consider the initial-valueproblem (IVP) (ν) t u u(ν) · u(ν) ν α u(ν) p (ν) θ (ν) e2 , θ (ν) u(ν) · θ (ν) θ (ν) 0,t(3.1) · u(ν) 0, (ν)u (x, 0) u0 (x), θ (ν) (x, 0) θ0 (x),where 0 α 1 and ν 0 are real parameters. When ν 0, (3.1) formally reduces to theIVP for the 2D Boussinesq equations with critical dissipation t u u · u p θ e2 , θ u · θ θ 0,t(3.2) · u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x),The local well-posedness of (3.1) and (3.2) follows from Kato [27]. We show that the solutionsof (3.1) converge to the corresponding ones of (3.2) with an explicit rate as ν 0. Moreprecisely, we have the following theorem.Theorem 3.1. Let ν 0 and α (0, 1]. Let σ 3. Consider (3.1) with (u0 , θ0 ) H σ (R2 ).Let T 0. Then there exists ν0 ν0 (T ) 0 such that, for 0 ν ν0 , (3.1) has aunique global solution satisfying (u(ν) , θ (ν) ) C([0, T ]; H σ (R2 )). In addition, for any2219
Nonlinearity 27 (2014) 2215J Wu and X Xu0 s σ 1 and 0 ν ν0 , the difference between (u(ν) , θ (ν) ) and the correspondingsolution (u, θ ) of (3.2) satisfies (u(ν) , θ (ν) ) (u, θ ) H s C(T ) ν,where C C(T ) is a constant dependent on T and (u, θ ) L ([0,T ];H s 1 ) only.Similar results can also be established for the 2D Boussinesq equations (κ) t u u(κ) · u(κ) u(κ) p (κ) θ (κ) e2 , θ (κ) u(κ) · θ (κ) κ β θ (κ) 0,t · u(κ) 0, (κ)u (x, 0) u0 (x),(3.3)θ (κ) (x, 0) θ0 (x),where κ 0 and 0 β 1 are real parameters. When κ 0, (3.3) formally reduces to theIVP for the critical Boussinesq–Navier–Stokes equations t u u · u u p θ e2 , θ u · θ 0,t(3.4) · u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x).We show that the solutions of (3.3) converge to the corresponding ones of (3.4) with an explicitrate as κ 0. More precisely, we have the following theorem.Theorem 3.2. Let κ 0 and β (0, 1]. Let σ 3. Consider (3.3) with (u0 , θ0 ) H σ (R2 ).Let T 0. Then there exists κ0 κ0 (T ) 0 such that, for 0 κ κ0 , (3.3) has aunique global solution satisfying (u(κ) , θ (κ) ) C([0, T ]; H σ (R2 )). In addition, for any0 s σ 1 and 0 κ κ0 , the difference between (u(κ) , θ (κ) ) and the correspondingsolution (u, θ ) of (3.4) satisfies (u(κ) , θ (κ) ) (u, θ ) H s C(T ) κ,where C C(T ) is a constant dependent on T and (u, θ ) L ([0,T ];H s 1 ) only.To prove theorem 3.1, we need a lemma assessing the global existence of classical solutionsto (3.2). It is obtained by combining the work by Hmidiet al [24] with the propagation ofregularity.Lemma 3.3. Assume that (u0 , θ0 ) H s (R2 ) with s (2, ). Then (3.2) has a unique globalsolution (u, θ ) satisfying, for any T 0,(u, θ ) C([0, T ]; H s (R2 )).(3.5)Proof of lemma 3.3. Since (u0 , θ0 ) H s (R2 ) with s 2, we have, for any q (2, ),1(R2 ) Ẇ 1,q (R2 ),u0 B ,10θ0 B ,1(R2 ) Lq (R2 ),01and B ,1denote inhomogeneous Besov spaces as defined in the appendix, andwhere B ,11,qẆ denotes a homogeneous Sobolev space. By theorem 1.1 of [24, p 422], (3.2) has a uniqueglobal solution (u, θ ) satisfying, for any T 0,101 1 ([0, T ]; Bq, Ẇ 1,q ),θ L ([0, T ]; B ,1 Lq ) L),u L ([0, T ]; B ,111 1 ([0, T ]; Bq, where L) is defined in the appendix. Since B ,1 L , we haveT u L dt .02220
Nonlinearity 27 (2014) 2215J Wu and X XuThe desired regularity (3.5) then follows from lemma 2.2. This completes the proof oflemma 3.3. (ν)σ2Proof of theorem 3.1. Since (u(ν)0 , θ0 ) H (R ) with σ 3, the local solution is guaranteedby lemma 2.2. To prove the global well-posedness, it suffices to obtain a global a priori boundfor (u(ν) , θ (ν) ) H σ . This is achieved through two steps. The first step is to compare (u(ν) , θ (ν) )with a solution (u, θ ) of (3.2) to obtain a bound for (u(ν) , θ (ν) ) H s for any s σ 1. Thedetails will be provided below. Since σ 1 2, the bound in the first step especially impliesthatT u(ν) L dt .0With this bound at our disposal, the second step is to use the regularity criterion in lemma 2.2to establish the global bound for (u(ν) , θ (ν) ) H σ .To implement the first step, we consider the differenceū u(ν) u,which satisfyθ̄ θ (ν) θ,p̄ p(ν) p, t ū u · ū ū · (u ū) ν α (u ū) p̄ θ̄ e2 , t θ̄ u · θ̄ ū · (θ θ̄) θ̄ 0, · ū 0, ū(x, 0) 0, θ̄ (x, 0) 0.(3.6)According to lemma 3.3, (u, θ ) C([0, T ]; H σ ). Now let 1 s σ 1. To estimate thesH s -norm of (ū, θ̄ ), we apply J s (I d ) 2 to (3.6) and then take the inner product with(J s ū, J s θ̄ ) to obtain1 d ū 2H s ν α/2 ū 2H s K11 K12 K13 K14 K15 ,(3.7)2 dt1 d(3.8) θ̄ 2H s 1/2 θ̄ 2H s K21 K22 K23 ,2 dtwhereK11 J s (u · ū) J s ū dx,K12 K13 J s (ū · ū) J s ū dx,K14 K15 ν α J s u J s ū dx,K21 K22 J s (ū · θ ) J s θ̄ dx,K23 J s (ū · u) J s ū dx,J s (θ̄ e2 ) J s ū dx,J s (u · θ̄) J s θ̄ dx,J s (ū · θ̄) J s θ̄ dx.Thanks to · u 0, Hölder’s inequality, a commutator estimate and Sobolev embedding, s ss J (u · ū) u · J ū J ū dx K11 J s (u · ū) u · J s ū L2 J s ū L2 C ( ū L J s u L2 u L J s ū L2 ) J s ū L2 C u H s ū 2H s .Similarly, K13 C ū 3H s .2221
Nonlinearity 27 (2014) 2215J Wu and X XuSince H s (R2 ) with s 1 is an algebra, K12 J s (ū · u) L2 J s ū L2 ū H s u H s J s ū L2 u H s 1 ū 2H s .By Hölder’s inequality, K14 θ̄ H s ū H s , K15 ν u H α s ū H s .Inserting the estimates above in (3.7), we findd ū H s C ū 2H s C u H s 1 ū H s ν u H α s θ̄ H s .dtK21 , K22 and K23 obey similar bounds as K11 , K12 and K13 , respectively. That is, K21 C u H s θ̄ 2H s , K22 C θ H s 1 ū H s θ̄ H s ,(3.9) K23 C ū H s θ̄ 2H s .Inserting these estimates in (3.8), we haved θ̄ H s C ū H s θ̄ H s C u H s θ̄ H s C θ H s 1 ū H s .(3.10)dtAdding (3.9) and (3.10) and setting Y (t) ū(t) H s θ̄(t) H s , we finddY ν u H α s C1 (1 u H s 1 θ H s 1 )Y C2 Y 2 ,dtwhere C1 and C2 are constants independent of ν and α. Since α (0, 1], s σ 1 and(u, θ ) C([0, T ]; H σ ), we apply lemma 2.1 to conclude that, if1ν0 T TC(1 u( ) H s 1 θ ( ) H s 1 ) d dτ8C2 0 u(τ ) H σ e 1 τthen, for 0 ν ν0 and 0 t T ,TY (t) 12ν u(τ ) H σ eC1 Tτ(1 u( ) H s 1 θ ( ) H s 1 ) d dτ.0This completes the proof of theorem 3.1. We remark that the proof of theorem 3.2 is similar and is thus omitted.4. Inviscid limits of general Boussinesq equationsThis section is concerned with the global regularity and inviscid limits of a general ddimensional (d-D) Boussinesq equations with dissipation given by a fractional Laplacian.Consider the IVP for the d-D Boussinesq equations α t u u · u ν u p θ ed , θ u · θ κ β θ 0,0t(4.1) · u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x),where ν 0, α 0, κ0 0 and β 0 are real parameters, and ed is the unit vector in thedirection of the last coordinate axis. κ0 0 is fixed and we study the limit as ν 0. Whenwe set ν 0, (4.1) formally reduces to the IVP for t u u · u p θ ed , θ u · θ κ β θ 0,t0(4.2) ·u 0, u(x, 0) u0 (x), θ (x, 0) θ0 (x).2222
Nonlinearity 27 (2014) 2215J Wu and X XuWe establish that, if (4.2) has a classical solution on the time interval [0, T ], then (4.1) with smallν 0 also has a classical solution on [0, T ] and the solution approaches the correspondingsolution of (4.2) as ν 0. More precisely, we have the following theorem.Theorem 4.1. Assume that (u0 , θ0 ) H σ (Rd ) with σ 2 d2 . Let T 0
Boussinesq equations with viscosity or thermal diffusivity in a suitable range also possess a classical solution on [0,T]. This result extends the work of Constantin on the Euler and the Navier–Stokes equations [13]. We defer the precise statement and the proof to section 4.
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