On Global Regularity of 2D Generalized Magnetohydrodynamic Equations
aa r X i v : . [ m a t h . A P ] F e b ON GLOBAL REGULARITY OF 2D GENERALIZEDMAGNETOHYDRODYNAMIC EQUATIONS
CHUONG V. TRAN, XINWEI YU, ZHICHUN ZHAI
Abstract.
In this article we study the global regularity of 2D generalized magnetohydrodynamicequations (2D GMHD), in which the dissipation terms are − ν ( −△ ) α u and − κ ( −△ ) β b . Weshow that smooth solutions are global in the following three cases: α > / , β >
1; 0 α < / , α + β > α > , β = 0. We also show that in the inviscid case ν = 0, if β >
1, then smoothsolutions are global as long as the direction of the magnetic field remains smooth enough. Introduction
Recent mathematical studies of fluid mechanics have found it beneficial to replace the Laplaceoperator △ , representing molecular diffusion, by fractional powers of −△ . For the magnetohydro-dynamic (MHD) equations, this practice results in the generalized MHD (GMHD) system u t + u · ∇ u = −∇ p + b · ∇ b − ν Λ α u, (1) b t + u · ∇ b = b · ∇ u − κ Λ β b, (2) ∇ · u = ∇ · b = 0 , (3)which is the subject of the present study. Here ν, κ, α, β ≥ −△ ) / is defined in termsof Fourier transform by c Λ f ( ξ ) = | ξ | b f ( ξ ) . (4)Equations (1–3) have been studied in some detail by Wu [28, 29] and Cao and Wu [3], with anemphasis on the issue of solution regularity.The generalization of diffusion in the above manner has been implemented to other fluid systems,including the Navier–Stokes, Boussinesq, and surface quasi-geostrophic equations (see e.g. [4], [5],[11], [13], [14], [15], [22]). Studying these generalized equations has enabled researchers to gaina deeper understanding of the strength and weaknesses of available mathematical methods andtechniques, and, in some cases, motivated and inspired the invention of new methods. An illustratingexample of the latter effect is the recent breakthroughs in the study of the surface quasi-geostrophicequations ([1], [7], [17], [18]).The problem of global well-posedness of the usual n -dimensional ( n D) MHD (or GMHD with α, β ≤
2) equations, where n ≥
3, is highly challenging for obvious reasons. One is that the MHDequations include the Navier-Stokes (or Euler when ν = 0) system as a special case (obtained bysetting the initial magnetic field to zero), for which the issue of regularity has not been resolved.Another is that the quadratic coupling between u and b can introduce additional technical diffi-culties, even though this coupling may actually have some regularizing effects (see below). For Date : Apr. 16, 2012.2000
Mathematics Subject Classification.
Key words and phrases.
Magnetohydrodynamics, Generalized diffusion, Global regularity. n = 2, this coupling invalidates the vorticity conservation, thereby becoming the main reason forthe unavailability of a proof of global regularity for the ideal dynamics. Similar (but probably moremanageable) situations arise when the 2D Euler equations are linearly coupled with the bouyancyequation in the Boussinesq system or have a linear forcing term ([7]).So far the best result for the global regularity of the n D GMHD equations (1–3) has beenderived in [30], where it has been proved that the system is globally regular as long as the followingconditions α >
12 + n , β > , α + β > n , (5)are satisfied. Note that for simplicity of presentation, the above conditions have been given inslightly stronger forms than the exact result in [30], where the dissipation terms are allowed tobe logarithmically weaker than − Λ α u and − Λ β b . Note also that for the case n = 3, conditionssimilar to (5) have been obtained in [31], with β > β > n >
3, the result (5) is unlikely to be improved using current mathematical techniques.The reason is that the global regularity for the n D generalized Navier-Stokes equations u t + u · ∇ u = −∇ p − Λ α u, ∇ · u = 0 . (6)is still unavailable for α < + n (See [25] for a proof of global regularity in the case of logarithmicallyweaker dissipation than − Λ n/ u ). On the other hand, when n = 2, the availability of globalregularity for the generalized Navier-Stokes equations (6) for all α > u or b guarantees that of the other and therefore of the system as awhole ([26]). Hence, global regularity could intuitively be possible with either ν = 0 or κ = 0 forsuitable conditions on β or α .In this article, we quantitatively confirm the above observations. More precisely, we show thatwhen n = 2, the condition α > + n is not needed for the global regularity of the system.In particular, we focus on the regime α < α < / , α + β > α > / , β >
1. We also prove global regularity forthe case α > , κ = 0, thereby removing the technical condition β >
0. Furthermore, we studythe inviscid case ν = 0, κ >
0, and show that when β >
1, the GMHD system is globallyregular as long as the magnetic lines are smooth enough. This result is consistent with numericaland experimental observations of the MHD dynamics, where the magnetic field appears to havethe effect of “suppressing” the appearance of small scales in the fluid (see e.g. [20]), and as aconsequence preventing the formation of singularities. Our finding is also consistent with a numberof mathematical results exhibiting the regularizing effect on the streamlines and vortex lines inNavier-Stokes and Euler dynamics (See e.g. [6], [9], [10], [27]).The rest of this article is organized as follows. In Section 2 we summarize the main results andgive a brief overview of the key ideas of their proofs. As these proofs use different methods for eachcase, we present them in separate sections. Section 3 features the proof for global regularity when α > / , β >
1. Sections 4 and 5 contain the proofs for the cases 0 α < / , α + β > α > , β = 0, respectively. In Section 6 we prove global regularity under the assumption on thesmoothness of magnetic lines.Throughout this paper, we will set κ = ν = 1 to simplify the presentation. It is a standardexercise to adjust various constants to accommodate other values of κ, ν , as long as both arepositive. We also identify the cases α = 0 and β = 0 with ν = 0 and κ = 0, respectively. LOBAL REGULARITY OF 2D GENERALIZED MHD 3 Main Results
Our first main result is the following global regularity theorem.
Theorem 1.
Consider the GMHD equations (1–3) in 2D. Assume ( u , b ) ∈ H k with k > . Thenthe system is globally regular for the following α, β : • α > / , β > ; • α < / , α + β > ; • α > , β = 0 . Remark 1.
Combining the above theorem with the main result in [30] , we see that the 2D GMHDsystem is globally regular for all α + β > except for α = 0 , β = 2 . Thus we have removed almostall technical conditions on α and β . The three cases will be proved using different methods, as different types of cancellation of the2D GMHD system will be exploited. More specifically, • for α > / , β >
1, we apply standard L -based energy method, taking advantage of thespecial cancellation that occurs for estimates in H . • for 0 α < / , α + β >
2, we derive a new non blow-up criterion in L p norm of thevorticity ω = ∇ ⊥ · u = − ∂ u + ∂ u and then show that this criterion is indeed satisfied. • for α > β = 0, we adapt the idea proposed in [21], carrying out a kind of “weaklynonlinear” energy estimate which takes advantage of the fact that in this case we have“almost” H a priori bound.Our second main result is the following theorem dealing with the case ν = 0 (for our purpose thisis the same as α = 0 since we do not impose any restriction on the size of the initial data). Theorem 2.
Consider the GMHD system (1–3) in 2D with α = 0 and β > . Assume ( u , b ) ∈ H k with k > . Then the system is globally regular if b b := b | b | ∈ L ∞ (cid:0) , T ; W , ∞ (cid:1) . Remark 2.
The condition on b b seems to be independent of the value of β , in the sense that thereis no β such that as soon as β > β , b b automatically belongs to L ∞ (cid:0) , T ; W , ∞ (cid:1) . Notation.
In the following we will use the standard function spaces L p , W k,p , H k whose normsare defined as k f k L p := (cid:18)Z R | f | p d x (cid:19) /p , k f k W k,p := X | α | = k k ∂ α f k pL p /p , k f k H k := k f k W k, with standard modifications for the case p = ∞ .3. Proof of Theorem 1 Case I: α > / , β > . In this section we prove the first case of Theorem 1. We apply standard L -based energy esti-mates. The key idea here is to carry out the H , H , H k estimates successively to explore possiblecancellations at each stage. We would like to mention that the cancellation at the H stage hasbeen observed before by several authors in the case β = 1 ([3],[21]). The general case β > CHUONG V. TRAN, XINWEI YU, ZHICHUN ZHAI H estimates ( L estimates for ω, j ).Lemma 1. ( H estimate) Consider the 2D GMHD equations (1–3), where α > and β > . Let ω = ∇ ⊥ · u = − ∂ u + ∂ u and j = ∇ ⊥ · b . Let u , b ∈ H . For fixed T > and < t < T , wehave k ω k L ( t ) + k j k L ( t ) + Z t (cid:16) k Λ α ω k L + (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L (cid:17) d τ C ( u , b , T ) . (7) Proof.
We first apply ∇ ⊥ · to the GMHD equations (1–3) to obtain the governing equations for thevorticity ω and the current j : ω t + u · ∇ ω = b · ∇ j − Λ α ω, (8) j t + u · ∇ j = b · ∇ ω + T ( ∇ u, ∇ b ) − Λ β j. (9)Here T ( ∇ u, ∇ b ) = 2 ∂ b ( ∂ u + ∂ u ) + 2 ∂ u ( ∂ b + ∂ b ) . (10)Note that T is bilinear in ∇ u, ∇ b and therefore for any k > (cid:12)(cid:12) ∂ k T ( ∇ u, ∇ b ) (cid:12)(cid:12) C k X m =0 (cid:12)(cid:12) ∇ m +1 u (cid:12)(cid:12) (cid:12)(cid:12) ∇ k − m +1 b (cid:12)(cid:12) (11)for some constant C depending only on k .Multiplying (8) and (9) by ω and j , respectively, integrating, and adding the resulting equationstogether we obtain12 dd t Z R2 (cid:0) ω + j (cid:1) d x = Z R T ( ∇ u, ∇ b ) j d x − Z R (Λ α ω ) d x − Z R (cid:0) Λ β j (cid:1) d x, (12)where we have used the following consequences of ∇ · u = ∇ · b = 0: Z R ( u · ∇ ω ) ω d x = 0; (13) Z R ( u · ∇ j ) j d x = 0; (14) Z R ( b · ∇ j ) ω d x + Z R ( b · ∇ ω ) j d x = 0 . (15)Note that all the terms involving derivatives of ω and j – the “worst” terms from energy estimatepoint of view – disappear.Now recall the standard energy conservation which can be obtained by multiplying (1) and (2)by u and b respectively, integrating, and applying the incompressibility condition (3):12 dd t Z R (cid:0) u + b (cid:1) d x + Z R h (Λ α u ) + (cid:0) Λ β b (cid:1) i d x = 0 . (16)This gives u ∈ L ∞ (cid:0) , T ; L (cid:1) ∩ L (0 , T ; H α ) , b ∈ L ∞ (cid:0) , T ; L (cid:1) ∩ L (cid:0) , T ; H β (cid:1) . (17)As β > b ∈ L (cid:0) , T ; H (cid:1) = ⇒ j ∈ L (cid:0) , T ; L (cid:1) . (18)On the other hand we have k Λ j k L C k b k aL (cid:13)(cid:13) Λ β j (cid:13)(cid:13) − aL (19) LOBAL REGULARITY OF 2D GENERALIZED MHD 5 for a = β − β + 1 . (20)Using Young’s inequality we obtain k Λ j k L a k b k L + (1 − a ) (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L = ⇒ (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L > − a k Λ j k L − a − a k b k L . (21)It is worth emphasizing that the above calculation remains valid even when a = 0, that is β = 1.This leads us todd t (cid:16) k ω k L + k j k L (cid:17) C Z R |∇ u | |∇ b | | j | d x − − a ) k Λ j k L + a (1 − a ) k b k L − k Λ α ω k L − (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L . (22)By H¨older’s inequality, the trilinear term satisfies Z R |∇ u | |∇ b | | j | d x k∇ u k L k∇ b k L k j k L . (23)Owing to the relations ∇ u = ∇ ( −△ ) − ∇ ⊥ ω and ∇ b = ∇ ( −△ ) − ∇ ⊥ j (24)we have, following standard Fourier multiplier theory (see e.g. [24]), k∇ u k L C k ω k L and k∇ b k L C k j k L (25)for some absolute constant C . It follows that Z R |∇ u | |∇ b | | j | d x C k ω k L k j k L . (26)Next, application of the Gagliardo-Nirenberg inequality k j k L C k j k / L k Λ j k / L (27)yields Z R |∇ u | |∇ b | | j | d x C k ω k L k j k L k Λ j k L C ( ε ) k j k L k ω k L + ε k Λ j k L , (28)where Young’s inequality has been used. Here ε is a small positive number that will be chosen later.Summarizing the above, we havedd t (cid:16) k ω k L + k j k L (cid:17) + k Λ α ω k L + (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L C ( ε ) k j k L k ω k L + Cε k Λ j k L − − a ) k Λ j k L + a (1 − a ) k b k L . (29)Taking ε small enough so that Cε < − a , we obtaindd t (cid:16) k ω k L + k j k L (cid:17) + (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L + k Λ α ω k L C ( ε ) k j k L k ω k L + a − a k b k L . (30)As k b k L is uniformly bounded in t , and k j k L ∈ L (0 , T ) ((17 – 18)), the proof is completed. (cid:3) CHUONG V. TRAN, XINWEI YU, ZHICHUN ZHAI
Remark 3.
Note that the above proof can be shortened by skipping the steps k Λ j k L k b k aL (cid:13)(cid:13) Λ β j (cid:13)(cid:13) − aL (31) and (cid:13)(cid:13) Λ β j (cid:13)(cid:13) L > − a k Λ j k L − a − a k b k L (32) and directly applying the Gagliardo-Nirenberg inequality k j k L k j k a L (cid:13)(cid:13) Λ β b (cid:13)(cid:13) a L (cid:13)(cid:13) Λ β j (cid:13)(cid:13) a L (33) for appropriate a , a , a , and then use Young’s inequality. However we choose to first reduce thegeneral situation β > to the particular one β = 1 to illustrate the following observation: Forour problem, to prove regularity for α > α , β > β using energy method, it suffices to do so for α = α , β = β . Such reduction significantly reduces the number of parameters in higher Sobolevnorm estimates and makes the presentation much more transparent, as we will see in the following H estimate. H estimates ( H estimates for ω, j ). With H estimates at hand, we can move on to H estimates. Differentiating (8–9) we reach( ∂ i ω ) t + u · ∇ ( ∂ i ω ) = − ( ∂ i u ) · ∇ ω + ( ∂ i b ) · ∇ j + b · ∇ ( ∂ i j ) − Λ α ( ∂ i ω ) (34)( ∂ i j ) t + u · ∇ ( ∂ i j ) = − ( ∂ i u ) · ∇ j + ( ∂ i b ) · ∇ ω + b · ∇ ( ∂ i ω ) + ∂ i ( T ( ∇ u, ∇ b )) − Λ β ( ∂ i j ) . (35)This gives the following integral relation:dd t Z R ( ∂ i ω ) + ( ∂ i j ) x = − Z R [( ∂ i u ) · ∇ ω ] ( ∂ i ω ) d x + Z R [( ∂ i b ) · ∇ j ] ( ∂ i ω ) d x − Z R [( ∂ i u ) · ∇ j ] ( ∂ i j ) d x + Z R [( ∂ i b ) · ∇ ω ] ( ∂ i j ) d x + Z R [ ∂ i ( T ( ∇ u, ∇ b ))] ( ∂ i j ) d x − Z R (Λ α ∂ i ω ) d x − Z R (cid:0) Λ β ∂ i j (cid:1) d x. (36)after taking advantage of ∇ · u = ∇ · b = 0.Summing up i = 1 ,
2, we reachdd t (cid:16) k∇ ω k L + k∇ j k L (cid:17) C ( I + I + I + I + I ) − k Λ α ∇ ω k L − (cid:13)(cid:13) Λ β ∇ j (cid:13)(cid:13) L (37)with C an absolute constant, and I = Z R |∇ u | |∇ ω | d x ; (38) I = Z R |∇ b | |∇ j | |∇ ω | d x ; (39) I = Z R |∇ u | |∇ j | d x ; (40) I = Z R |∇ b | |∇ ω | |∇ j | d x ; (41) I = Z R (cid:2)(cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ b | + |∇ u | (cid:12)(cid:12) ∇ b (cid:12)(cid:12)(cid:3) |∇ j | d x. (42) LOBAL REGULARITY OF 2D GENERALIZED MHD 7
We estimate these quantities one by one. As discussed in Remark 3, we only need to carry out theestimates for the case α = 1 / , β = 1.There are four different cases ( I and I are identical). • Estimating I = R R |∇ u | |∇ ω | d x .First, by H¨older’s inequality we have I k∇ u k L k∇ ω k L C k ω k L k∇ ω k L . (43)Consider the following Gagliardo-Nirenberg inequalities. k∇ ω k L C (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) / L (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) / L ; (44) k∇ ω k L C k∇ ω k / L (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) / L ; (45) k ω k L C k ω k / L (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) / L . (46)Equations (44) and (45) imply k∇ ω k L = k∇ ω k / L k∇ ω k / L C (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) / L k∇ ω k / L (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) / L . (47)Now (46) and (47) gives I C k ω k L k∇ ω k L C k ω k / L (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) / L k∇ ω k / L (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) / L . (48)Applying Young’s inequality we get I C ( ε ) k ω k L (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) L k∇ ω k L + ε (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) L . (49)Here ε can be taken as small as we want and will be specified later. • Estimating I = I = R R |∇ b | |∇ j | |∇ ω | d x .Using H¨older’s inequality we have Z R |∇ b | |∇ j | |∇ ω | d x k∇ b k L k∇ j k L k∇ ω k L C k j k L k∇ j k L k∇ ω k L . (50)Applying the Gagliardo-Nirenberg inequalities k j k L C k j k / L k∇ j k / L ; k∇ j k L C k∇ j k / L k Λ ∇ j k / L (51)yields Z R |∇ b | |∇ j | |∇ ω | d x C k j k / L k∇ j k L k Λ ∇ j k / L k∇ ω k L . (52)Applying Young’s inequality further yields Z R |∇ b | |∇ j | |∇ ω | d x C ( ε ) k j k L + k∇ j k L k∇ ω k L + ε k Λ ∇ j k L . (53) • Estimating I = R R |∇ u | |∇ j | d x .Using H¨older’s inequality we have Z R |∇ u | |∇ j | d x k∇ u k L k∇ j k L C k ω k L k∇ j k L . (54) CHUONG V. TRAN, XINWEI YU, ZHICHUN ZHAI
Now using the second Gagliardo-Nirenberg inequality in (51) and Young’s inequality we get I C k ω k L k∇ j k L k Λ ∇ j k L C ( ε ) k ω k L k∇ j k L + ε k Λ ∇ j k L . (55) • Estimating I = R R (cid:2)(cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ b | + |∇ u | (cid:12)(cid:12) ∇ b (cid:12)(cid:12)(cid:3) |∇ j | d x .We write I = I + I := Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ b | |∇ j | d x + Z R |∇ u | (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ j | d x. (56)It is clear that I can be estimated similar to I while I can be estimated similar to I . Remark 4.
We would like to emphasize that the assumption α > / is only needed for theestimation of I . The estimates I − I only require α > , β > . Putting the above results together, we havedd t (cid:16) k∇ ω k L + k∇ j k L (cid:17) C ( ε ) (cid:20) k ω k L (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) L + k ω k L + k∇ j k L + 1 (cid:21) (cid:16) k∇ ω k L + k∇ j k L (cid:17) + C ( ε ) k j k L − (cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) L − k Λ ∇ j k L + Cε (cid:18)(cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) L + k Λ ∇ j k L (cid:19) . (57)Taking ε small enough so that Cε < t (cid:16) k∇ ω k L + k∇ j k L (cid:17) C ( ε ) (cid:20) k ω k L (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) L + k ω k L + k∇ j k L + 1 (cid:21) (cid:16) k∇ ω k L + k∇ j k L (cid:17) + C ( ε ) k j k L − (cid:18)(cid:13)(cid:13)(cid:13) Λ / ∇ ω (cid:13)(cid:13)(cid:13) L + k Λ ∇ j k L (cid:19) . (58)Recall that (cid:13)(cid:13)(cid:13) Λ / ω (cid:13)(cid:13)(cid:13) L , k∇ j k L ∈ L (0 , T ) ; k ω k L , k j k L ∈ L ∞ (0 , T ) (59)thanks to the H estimate. This, together with (58), implies ∇ ω, ∇ j ∈ L ∞ (cid:0) , T ; L (cid:1) . (60)Combining with the H estimate, we have the following H estimate: k ω k H + k j k H ∈ L ∞ (0 , T ) . (61)3.3. H k estimates. An argument which by now is standard (see for example [21]) generalizes theclassical BKM-type blow-up criterion ([2]) toThe MHD system stays regular beyond T if and only if Z T ( k ω k BMO + k j k BMO ) d t < ∞ . (62)Using the embedding H ֒ −→ BMO (63)in 2D, we see that k ω k H + k j k H ∈ L ∞ (0 , T ) = ⇒ k ω k BMO + k j k BMO ∈ L ∞ (0 , T ) (64)and consequently all H k norms are bounded. This completes the proof of the first case. LOBAL REGULARITY OF 2D GENERALIZED MHD 9 Proof of Theorem 1 Case II: α < / , α + β > . To prove global regularity in this case, we first derive a blow-up criterion in k ω k L p for appropriate p , then obtain a priori estimate for k ω k L p . Note that in this case we have β > H β ֒ −→ L ∞ (65)in 2D already gives j ∈ L (0 , T ; L ∞ ) ֒ −→ L (0 , T ; BMO). Lemma 2.
Assume < α < / , β > . The GMHD system (1–3) is regular if ω ∈ L p for any p > α .Proof. As we have β >
1, we already have the following H estimates thanks to Lemma 1: ω ∈ L ∞ (cid:0) , T ; L (cid:1) ∩ L (0 , T ; H α ) ; j ∈ L ∞ (cid:0) , T ; L (cid:1) ∩ L (cid:0) , T ; H β (cid:1) . (66)Now arguing similarly as in Sections 3.2 and 3.3, we see that all we need to do is to bound I − I as defined in (38–42). Furthermore, we note that the estimates for I − I can be done similarly tothat in Section 3.2, as explained in Remark 4. The only estimate that needs to be done differentlyis that of I = R R | ω | |∇ ω | d x .For that purpose, we first apply H¨older’s inequality to I to obtain Z R | ω | |∇ ω | d x k ω k L p k∇ ω k L q (67)for p , q satisfy p > α , p + 1 q = 1 . (68)Next we use the following Gagliardo-Nirenberg type inequalities: k∇ ω k L q C k Λ α ω k ξL k Λ α ∇ ω k − ξL with ξ = α − p = α (cid:18) − p α (cid:19) ; (69) k∇ ω k L q C k∇ ω k ηL k Λ α ∇ ω k − ηL with η = 1 − p α . (70)Note that as long as p > α both ξ, η ∈ (0 , a = α α (cid:18) − p α (cid:19) , (71)which satisfies 0 < a < / < α < / p > /α , we have k∇ ω k L q = k∇ ω k / (1+ α ) L q k∇ ω k α/ (1+ α ) L q C k Λ α ω k aL k∇ ω k aL k Λ α ∇ ω k − aL . (72)Next we apply the following Gagliardo-Nirenberg inequality k ω k L p C k ω k − aL p k Λ α ∇ ω k aL , (73)where a is given by (71) and p < p . The exact value of p can be written down but what is importanthere is that p > α , as can be seen from the following manipulation of the scaling relation: − p = (1 − a ) (cid:18) − p (cid:19) + 2 aα = ⇒ − p = (1 − a ) (cid:18) − p (cid:19) + aα. (74) Writing (71) as a = α (cid:16) α − p (cid:17) and then adding α to both sides of (74), we reach α − p = 1 − α/ ( α + 1)1 − a (cid:18) α − p (cid:19) . (75)Recalling α < /
2, we see that α − /p > α − /p > I can be bounded as I k ω k L p k∇ ω k L q C k ω k − aL p (cid:16) k Λ α ω k aL k∇ ω k aL k Λ α ∇ ω k − aL (cid:17) C ( ε ) k ω k (1 − a ) /aL p k Λ α ω k L k∇ ω k L + ε k Λ α ∇ ω k L . (76)Now it is clear that once k ω k L p ∈ L ∞ (0 , T ), we can obtain H estimate as in Section 3.2, andglobal regularity follows as in Section 3.3.Finally, if k ω k L q is bounded for some q > α >
2, then together with the H estimate ω ∈ L ∞ (cid:0) , T ; L (cid:1) we see that k ω k L r ∈ L ∞ (0 , T ) ∀ r ∈ [2 , q ] . (77)Now we can simply take p = q in the above inequalities, then since p < p we have the uniformboundedness of k ω k L p and global regularity follows. (cid:3) Remark 5.
The case α = 0 (which we identify with the case ν = 0 ) is trivial. By our assumption α + β > we have β > , which gives ∇ j ∈ L (0 , T ; L ∞ ) . This result, together with the vorticityequation ω t + u · ∇ ω = b · ∇ j, (78) implies ω ∈ L ∞ (0 , T ; L ∞ ) . Global regularity then follows from the BKM type criterion in [2] . In light of Lemma 2, all we need to do is to show that when 2 α + β >
2, there is indeed p > α such that k ω k L p remains uniformly bounded over (0 , T ).Recall the equation for ω : ω t + u · ∇ ω = b · ∇ j − Λ α ω. (79)Multiply both sides by p | ω | p − ω and integrate we reachdd t Z R | ω | p d x p Z R | b | |∇ j | | ω | p − d x − p Z R (Λ α ω ) | ω | p − ω d x. (80)after taking advantage of ∇ · u = 0.For the dissipation term, it is well-known that Z R (Λ α ω ) | ω | p − ω d x > . (81)This is originally proved in [23], and has later been refined in [8], [16].Taking into account the above “positivity” property and using H¨older’s inequality, we obtaindd t k ω k L p k b · ∇ j k L p k b k L ∞ k∇ j k L p . (82)Now as β >
1, we have H estimate as in 3.1. In particular we have j ∈ L (cid:0) , T ; H β (cid:1) . (83)Sobolev embedding then gives j ∈ L (cid:0) , T ; H β (cid:1) = ⇒ ∇ j ∈ L (0 , T ; L p ) and b ∈ L (0 , T ; L ∞ ) . (84) LOBAL REGULARITY OF 2D GENERALIZED MHD 11 with p > α satisfying p − β when β < , and p < ∞ when β > . (85)As α + β >
2, such p exists. Now we have k ω k L p k ω k L p + Z t k b k L ∞ k∇ j k L p d τ k ω k L p + k b k L (0 ,T ; L ∞ ) k∇ j k L (0 ,T ; L p ) C ( ω , T ) . (86)Therefore k ω k L p ∈ L ∞ (0 , T ) and global regularity follows from Lemma 2.5. Proof of Theorem 1 Case III: α > , β = 0 . In this section we prove global regularity in the case α > , β = 0. As we identify β = 0 with κ = 0, the GMHD equations now reads u t + u · ∇ u = −∇ p + b · ∇ b − Λ α u, (87) b t + u · ∇ b = b · ∇ u, (88) ∇ · u = ∇ · b = 0 . (89)In what follows we will only present the proof for the case α = 2 , β = 0. The case α > α > ω ∈ L (0 , T ; L ∞ ). Thisleads to a priori H bounds which are sufficient to prove a priori H bounds.We will show that when α >
2, the H norms of ω and j must stay finite for any T >
0. Once thisis proved, Sobolev embedding immediately gives the finiteness of k ω k L ∞ and k j k L ∞ and regularityfollows. The H bound is proved by contradiction: Assume lim sup t ր T k ω k H + k j k H = ∞ forsome finite time T >
0. The idea is to start from a time T close enough to T and show thatunder such assumption k ω k H + k j k H remains uniformly bounded for T < t < T , thus reachinga contradiction.First observe that in this case, energy conservation gives u, b ∈ L ∞ (cid:0) , T ; L (cid:1) , △ u ∈ L (cid:0) , T ; L (cid:1) = ⇒ ∇ u, ω ∈ L (0 , T ; BMO) ֒ −→ L (0 , T ; BMO) . (90)5.1. H Estimates.
Similar to Section 3.1, we have12 dd t (cid:16) k ω k L + k j k L (cid:17) + k△ ω k L (cid:12)(cid:12)(cid:12)(cid:12)Z R T ( ∇ u, ∇ b ) j d x (cid:12)(cid:12)(cid:12)(cid:12) (91)Recalling (10) T ( ∇ u, ∇ b ) = 2 ∂ b ( ∂ u + ∂ u ) + 2 ∂ u ( ∂ b + ∂ b ) (92)and using k∇ b k L C k j k L , (93)we have (cid:12)(cid:12)(cid:12)(cid:12)Z R jT ( ∇ u, ∇ b ) d x (cid:12)(cid:12)(cid:12)(cid:12) C k∇ u k L ∞ k j k L . (94)This gives dd t (cid:16) k ω k L + k j k L (cid:17) + 2 k△ ω k L C k∇ u k L ∞ (cid:16) k ω k L + k j k L (cid:17) . (95) Here we make use of the following Gronwall-type inequality, which is a variant of the standardGronwall’s inequality as presented in [12], Appendix B.j.
Lemma 3.
Let η ( · ) be a nonnegative, absolutely continuous function on [0 , T ] , which satisfies fora.e. t the inequality η ′ ( t ) + ψ ( t ) φ ( t ) η ( t ) , (96) where φ ( t ) and ψ ( t ) are nonnegative, summable functions on [0 , T ] . Then η ( t ) + Z t ψ ( τ ) d τ η (0) exp (cid:20)Z t φ ( τ ) d τ (cid:21) . (97) Proof.
The proof follows the same idea as that presented in [12] and is omitted. (cid:3)
Taking η := k ω k L + k j k L and ψ := 2 k△ ω k L in Lemma 3, then integrating from T to t , weobtain Z tT k△ ω k L d τ k ω k L + k j k L + Z tT k△ ω k L d τ (cid:16) k ω k L + k j k L (cid:17) exp (cid:20) C Z tT k∇ u k L ∞ ( τ ) d τ (cid:21) . (98)Here T ∈ (0 , T ) will be fixed later and we denote ω := ω ( · , T ) , j := j ( · , T ).Now applying the logarithmic inequality (see e.g. [19]) k∇ u k L ∞ C (cid:16) k u k L + k ω k BMO (cid:16) (cid:16) k ω k H + k j k H (cid:17)(cid:17)(cid:17) (99)and setting M ( t ) := max τ ∈ ( T ,t ) (cid:16) k ω k H + k j k H (cid:17) ( τ ) (100)we reach Z tT k△ ω k L d τ (cid:16) k ω k L + k j k L (cid:17) exp [ C (1 + k u k L )] exp (cid:20) C (cid:18)Z tT k ω k BMO d τ (cid:19) (1 + log (1 + M ( t ))) (cid:21) . (101)Note that thanks to the energy estimate k u k L k u (0) k L so exp ( C (1 + k u k L )) is bounded byan constant independent of T .As k ω k BMO ∈ L ( T , T ), we can take T close enough to T so that C Z tT k ω k BMO d τ δ (102)for some small positive number δ to be fixed later. With such choice of T we have Z tT k△ ω k L d τ C ( T ) (1 + M ( t )) δ . (103)Now H¨older’s inequality gives Z tT k△ ω k L d τ C ( T ) (1 + M ( t )) δ . (104)Before proceeding, we fix T by the following requirements: C Z tT k ω k BMO d τ δ, log(1 + M ( T )) > . (105)At the end of Section 5.2 we will show that δ can be taken as 1 / LOBAL REGULARITY OF 2D GENERALIZED MHD 13 H estimate ( H estimate for ω, j ). In this subsection we prove the uniform boundednessof M ( t ) for all T < t < T , thus reaching contradiction.Let ∂ denote any double partial derivative (such as ∂ , ∂ etc.). Taking ∂ of (8) and (9)and multiplying the resulting equations by ∂ ω and ∂ j respectively, we reach, after using ∇ · u = ∇ · b = 0, 12 dd t Z R h(cid:0) ∂ ω (cid:1) + (cid:0) ∂ j (cid:1) i d x A + B + C + D + E − Z R (cid:0) △ ∂ ω (cid:1) d x, (106)with A = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) ∂ ( u · ∇ ω ) − u · ∇ ∂ ω (cid:3) (cid:0) ∂ ω (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ ω | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x + Z R |∇ u | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x ; (107) B = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) ∂ ( b · ∇ j ) − b · ∇ ∂ j (cid:3) (cid:0) ∂ ω (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x + Z R |∇ b | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x ; (108) C = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) ∂ ( u · ∇ j ) − u · ∇ (cid:0) ∂ j (cid:1)(cid:3) (cid:0) ∂ j (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R |∇ u | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x ; (109) D = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) ∂ ( b · ∇ ω ) − b · ∇ (cid:0) ∂ ω (cid:1)(cid:3) (cid:0) ∂ j (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ ω | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R |∇ b | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) (cid:12)(cid:12) ∇ j | d x ; (110) E = (cid:12)(cid:12)(cid:12)(cid:12)Z R ∂ T ( ∇ u, ∇ b ) (cid:0) ∂ j (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ b | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) (cid:12)(cid:12) ∇ b (cid:12)(cid:12) (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R |∇ u | (cid:12)(cid:12) ∇ b (cid:12)(cid:12) (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x. (111)Adding up all such partial derivatives, we obtaindd t (cid:16)(cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L (cid:17) C ( I + I + I + I + I + I ) − (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (112) with I = Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ ω | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x ; (113) I = Z R |∇ u | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x + Z R |∇ u | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R |∇ u | (cid:12)(cid:12) ∇ b (cid:12)(cid:12) (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x ; (114) I = Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x ; (115) I = Z R |∇ b | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x + Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ b | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x ; (116) I = Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x ; (117) I = Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ ω | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x + Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) (cid:12)(cid:12) ∇ b (cid:12)(cid:12) (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x. (118)We remark that the integrals in each I k can be estimated similarly, therefore in the following weonly show how to estimate the first integral in each I k . • I . For I we write I (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k∇ ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C k∇ ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C k u k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (119)where we have used the following Gagliardo-Nirenberg inequality k∇ ω k L C k u k / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L . (120)Now by Young’s inequality we have, after using k u k L k u k L , I C ( ε ) k u k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + ε (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C ( ε ) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + ε (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (121)with ε as small as necessary. • I . We have Z R |∇ u | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x k∇ u k L ∞ (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C (cid:16) k u k L + k ω k BMO (cid:16) (cid:16) k ω k H + k j k H (cid:17)(cid:17)(cid:17) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C (cid:16) k ω k BMO (cid:16) (cid:16) k ω k H + k j k H (cid:17)(cid:17)(cid:17) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (122)where we have used the logarithmic inequality (99). • I . We have Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x (cid:13)(cid:13) ∇ b (cid:13)(cid:13) L k∇ j k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C k∇ j k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (123)where we have used the following Gagliardo-Nirenberg inequality k∇ j k L C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L . (124) LOBAL REGULARITY OF 2D GENERALIZED MHD 15
As a consequence (recall the definition of M ( t ) in (100)) I C (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L M ( t ) / . (125)Here we have used the energy conservation k b k L k b k L + k u k L . • I . We have Z R |∇ b | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) (cid:12)(cid:12) ∇ ω (cid:12)(cid:12) d x k∇ b k L ∞ (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , (126)where we have used the following Gagliardo-Nirenberg inequality k∇ b k L ∞ C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L . (127)Therefore I C (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L M ( t ) / . (128) • I . We have I = Z R (cid:12)(cid:12) ∇ u (cid:12)(cid:12) |∇ j | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k∇ j k L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L C k u k / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L , (129)where we have used the following Gagliardo-Nirenberg inequalities (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L C k u k / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L ; k∇ j k L C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L . (130)Hence I C (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L M ( t ) / C (cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) M ( t ) / . (131) • I . We have Z R (cid:12)(cid:12) ∇ b (cid:12)(cid:12) |∇ ω | (cid:12)(cid:12) ∇ j (cid:12)(cid:12) d x (cid:13)(cid:13) ∇ b (cid:13)(cid:13) L k∇ ω k L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L C k∇ j k L k∇ ω k L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L k b k / L k u k / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L , (132)where we have used the following Gagliardo-Nirenberg inequalities k∇ ω k L C k u k / L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L ; k∇ j k L C k b k / L (cid:13)(cid:13) ∇ j (cid:13)(cid:13) / L . (133)Hence I C (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) / L M ( t ) / C (cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) M ( t ) / . (134)Summarizing, we havedd t (cid:16)(cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L (cid:17) C ( T ) h M ( t ) + (cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) M ( t ) / + k ω k BMO M ( t ) log (1 + M ( t ))] . (135) Using our assumption on T (105) and the monotonicity of M ( t ), we have log (1 + M ( t )) > t (cid:16)(cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ j (cid:13)(cid:13) L (cid:17) C ( T ) h(cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) M ( t ) / + (1 + k ω k BMO ) M ( t ) log (1 + M ( t ))] . (136)Integrating, we have M ( t ) C ( T ) (cid:20) M + (cid:18)Z tT (cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) d τ (cid:19) M ( t ) / + Z tT [(1 + k ω k BMO ) M ( τ ) log (1 + M ( τ ))] d τ (cid:21) , (137)with M := k ω k H ( T ) + k j k H ( T ).Now taking δ = 1 /
24, we have Z tT (cid:0) (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L (cid:1) d τ C ( T ) (1 + M ( t )) / , (138)which leads to M ( t ) C ( T ) h M + M ( t ) / (1 + M ( t )) / + Z tT [(1 + k ω k BMO ) M ( τ ) log (1 + M ( τ ))] d τ (cid:21) . (139)This in turn gives1 + M ( t ) C ( T ) h (1 + M ) + (1 + M ( t )) / + Z tT [(1 + k ω k BMO ) (1 + M ( τ )) log (1 + M ( τ ))] d τ (cid:21) . (140)Now we set N ( t ) := (1 + M ( t )) / , N := (1 + M ) / and divide both sides by (1 + M ( t )) / ,using the monotonicity of M ( t ) we reach N ( t ) C ( T ) (cid:20) (1 + N ) + Z tT (1 + k ω k BMO ) N ( τ ) log ( N ( τ )) d τ (cid:21) . (141)Application of the standard Gronwall’s inequality now gives the following bound of NN ( t ) [ C ( T ) (1 + N )] exp h C ( T ) R tT ( k ω k BMO ) d τ i , (142)which gives M ( t ) [ C ( T ) (1 + N )]
24 exp h C ( T ) R tT ( k ω k BMO ) d τ i . (143)Since R tT k ω k BMO ( τ ) d τ remains bounded as t ր T , (143) contradicts our assumption that M ( t ) ր∞ as t ր T and ends the proof. LOBAL REGULARITY OF 2D GENERALIZED MHD 17 H k estimate. As we have already proved that the H norms of ω and j have to remainbounded as t ր T , thanks to the embedding H ֒ −→ L ∞ in R , we have ω, j ∈ L ∞ (0 , T ; L ∞ ) (144)as a result of the argument in 5.1 and 5.2. The H k estimate and global regularity is now a simpleconsequence of the BKM-type criterion in [2].6. Global regularity when the magnetic lines are smooth
This section proves Theorem 2, which states that the system u t + u · ∇ u = −∇ p + b · ∇ b, (145) b t + u · ∇ b = b · ∇ u − Λ β b, (146) ∇ · u = ∇ · b = 0 , (147)with β > u , b ) ∈ H k for some k >
2, is globally regular if b b := b | b | ∈ L ∞ (cid:0) , T ; W , ∞ (cid:1) . Proof. As β >
1, following Lemma 1 we already have H estimate which in particular gives j ∈ L (cid:0) , T ; H β (cid:1) ֒ −→ L (0 , T ; L ∞ ) (148)since H β ֒ −→ L ∞ . Thanks to the BKM-type criteria in [2], all we need to prove is that ω ∈ L (0 , T ; L ∞ ).For a proof of ω ∈ L (0 , T ; L ∞ ), let us examine the vorticity equation ω t + u · ∇ ω = ∇ ⊥ · ( b · ∇ b ) , (149)where the “forcing” term has been given in its raw form instead of b · ∇ j for the very purpose ofthis proof. By writing b = b b | b | (150)and using the divergence free condition ∇ · b = 0, we have b b · ∇ | b | = − (cid:16) ∇ · b b (cid:17) | b | . (151)It follows that b · ∇ b = | b | hb b · ∇ (cid:16)b b | b | (cid:17)i = hb b · ∇ b b − (cid:16) ∇ · b b (cid:17) b b i | b | . (152)Therefore the vorticity equation can be written as ω t + u · ∇ ω = ∇ ⊥ · nhb b · ∇ b b − (cid:16) ∇ · b b (cid:17) b b i | b | o = A ( x, t ) | b | + B ( x, t ) · (cid:0) b · ∇ ⊥ b (cid:1) , (153)where A ( x, t ) = ∇ ⊥ · hb b · ∇ b b − (cid:16) ∇ · b b (cid:17) b b i , B ( x, t ) = b b · ∇ b b − (cid:16) ∇ · b b (cid:17) b b. (154)As b b ∈ W , ∞ by our assumption, we readily deduce that A ( x, t ) , B ( x, t ) ∈ L ∞ (0 , T ; L ∞ ) . (155)Now since β >
1, the earlier estimates in H mean j ∈ L (cid:0) , T ; H β (cid:1) = ⇒ ∇ b ∈ L (cid:0) , T ; H β (cid:1) = ⇒ ∇ b ∈ L (0 , T ; L ∞ ) . (156)It follows that | b | , b · ∇ ⊥ b ∈ L (0 , T ; L ∞ ) . (157) Putting things together, we see that ω t + u · ∇ ω = F ( x, t ) := A ( x, t ) | b | + B ( x, t ) · (cid:0) b · ∇ ⊥ b (cid:1) , (158)with F ( x, t ) ∈ L (0 , T ; L ∞ ). Since we are dealing with smooth solutions here, this immediatelyleads to ω ∈ L ∞ (0 , T ; L ∞ ) ֒ −→ L (0 , T ; L ∞ ) (159)and the proof is completed. (cid:3) Remark 6.
For solutions not smooth enough, we can argue as follows. First note that j ∈ L (cid:0) , T ; H β (cid:1) implies ∇ b ∈ L (0 , T ; L q ) for any q , and furthermore k∇ b k L (0 ,T ; L q ) is uniformlybounded in q . Consequently F ∈ L (0 , T ; L q ) for any q with uniformly bounded norms. Now wemultiply the equation by | ω | p − ω and integrate. After simplification we get dd t k ω k pL p p (cid:12)(cid:12)(cid:12)(cid:12)Z R F ( x, t ) | ω | p − ω d x (cid:12)(cid:12)(cid:12)(cid:12) p k F k L p k ω k p − L p , (160) which implies dd t k ω k L p k F k L p . (161) This gives a uniform bound on k ω k L p and consequently a bound on k ω k L ∞ . Acknowledgment . X. Yu and Z. Zhai are supported by a grant from NSERC and the Startup grantfrom Faculty of Science of University of Alberta. The authors would like to thank the anonymousreferee for the valuable comments and suggestions.
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Chuong V. Tran: School of Mathematics and Statistics, University of St. Andrews, St AndrewsKY16 9SS, United Kingdom
E-mail address : [email protected] Xinwei Yu and Zhichun Zhai: Department of Mathematical and Statistical Sciences, University ofAlberta, Edmonton, AB, T6G 2G1, Canada
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