Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics
aa r X i v : . [ m a t h . A P ] S e p DETERMINING WAVENUMBERS FOR THE INCOMPRESSIBLEHALL-MAGNETO-HYDRODYNAMICS
HAN LIU
Abstract.
Using Littlewood-Paley theory, one formulates the determiningwavenumbers for the Hall-MHD system, defined for each individual solution( u, b ). It is shown that the long time behaviour of strong solutions is almostfinite dimensional as the wavenumbers are bounded in certain average senses.KEY WORDS: Hall-MHD system; determining modesCLASSIFICATION CODE: 35Q35, 35Q85, 37L30 Introduction
This paper deals with the finite dimensionality of solutions to the incompressibleHall-magneto-hydrodynamics (Hall-MHD) system, written as follows u t + ( u · ∇ ) u − ( b · ∇ ) b + ∇ p = ν ∆ u + f, (1.1) b t + ( u · ∇ ) b − ( b · ∇ ) u + η ∇ × (( ∇ × b ) × b ) = µ ∆ b, (1.2) ∇ · u = 0 , ∇ · b = 0 , t ∈ R + , x ∈ T . (1.3)The above system describes the evolution of a system consisting of a magneticfield b, electrons and ions, whose collective motion under b can be approximatedas an electrically conducting fluid with velocity field u. The focus here is primarilythe visco-resistive case, corresponding to positive fluid viscosity ν and magneticresistivity µ. The external forcing term f , which one assumes to have zero mean,renders system (1.1)-(1.3) inhomogeneous. The Hall-MHD system is derived usinggeneralized Ohm’s law which takes into account the effect of the electric current onthe Lorentz force, neglected in the derivation of the MHD equations. The resultingextra Hall term η ( ∇ × (( ∇ × b ) × b )) , distinguishing system (1.1)-(1.3) from theconventional MHD system, becomes significant in the case of large magnetic shear.The coefficient η here is proportional to the ion skin length.The Hall-MHD system has a wide range of applications including modelling solarwinds, designing magnetic confinement devices for fusion reactors and interpretingthe origin of the geomagnetic field. It is believed to be an essential model for mag-netic reconnection, an intriguing phenomenon frequently observed in space plasmas.Over the past decade, the Hall-MHD system has received more attentions from themathematical community. Acheritogaray, Degond, Frouvelle and Liu [1] rigorouslyderived the system and established global existence of weak solutions on periodicdomains. Chae, Degond and Liu [5] proved existence of global weak solutions in R as well as that of local smooth solutions. Chae and Lee [6] obtained blow-upcriteria and small data global strong solutions. Evidences of ill-posedness can befound in [9, 18, 36]. As for the properties of solutions, temporal decay estimates in The work of the authors was partially supported by NSF Grant DMS–1108864. energy spaces are due to Chae and Schonbek [12]. For more mathematical resultson the Hall-MHD system, e.g., well-posedness results and regularity criteria, pleasesee [2, 15, 16, 17, 21, 22, 35, 39, 44, 45, 46, 47, 48, 49].In the case of b ≡ , system (1.1)-(1.3) reduces to the Navier-Stokes equations(NSE), for which the finite dimensional behaviour of solutions has been extensivelystudied. As alluded in Kolmogorov’s 1941 phenomenological theory [38], a turbulentflow should have a finite number of degrees of freedom. The first mathematicalresult in this direction, due to Foia¸s and Prodi [26], stated that the higher Fouriermodes of a solution to the 2D NSE are controlled by the lower modes asymptoticallyas time goes to infinity. More precisely, if a certain finite number of Fourier modesof a solution share the same long time behaviour with those of another solution,then the remaining infinitely many Fourier modes of the two solutions also exhibitthe same long time behaviour. Thus, the notion of “determining modes” arisesnaturally. For the 2D NSE, estimates of the number of the determining modes wereobtained by Foia¸s, Manley, Temam and Treve [25] in terms of the Grashof number,and later improved by Jones and Titi [37], whereas Constantin, Foia¸s, Manley andTemam [13] estimated the number of determining modes for the 3D NSE assumingthe uniform boundedness of solutions in H . For more details concerning the studyof finite dimensionality of the NSE flow, readers are referred to [14, 23, 24, 27, 28,29, 43].Motivated by the work of Cheskidov, Dai and Kavlie [11] where a time-dependentdetermining wavenumber was introduced to estimate the number of determiningmodes for weak solutions to the 3D NSE in an average sense, this paper aims toadapt the idea therein to the study of the Hall-MHD system. In particular, as finitedimensionality of the closely related MHD system has been investigated by Edenand Libin in [20], one is curious if such results can also be obtained for the Hall-MHD system, which differs nontrivially from the MHD system in many aspects, asillustrated in [6, 9, 36].One introduces the determining wavenumbers for an individual weak solution( u, b ) to system (1.1)-(1.3). Let κ := min { µ, ν, η − µ } , r ∈ (2 ,
3) and δ > . Let c r be a constant depending only on r. The determining wavenumbers correspondingto u and b are defined as follows.Λ u ( t ) =: min (cid:8) λ q : λ − r p k u p k r < c r κ, ∀ p > q ; λ − r q k u ≤ q k r < c r κ, q ∈ N (cid:9) , (1.4) Λ b ( t ) =: min (cid:8) λ q : λ δp − q k b p k ∞ < c r κ, ∀ p > q ; k b ≤ q k ∞ < c r κ, q ∈ N (cid:9) , (1.5)where λ q = 2 q and u q = ∆ q u, the q -th Littlewood-Paley projection of u, to bedefined in Section 2. One notices that in both definitions the conditions on the highmodes resemble the ones in the definitions of the dissipation wavenumbers foundin [10, 15]. It is noteworthy that the dissipation wavenumber, which separates thedissipation range from the inertial range of turbulent flows, has been utilized toestablish improved regularity criteria for various fluid models in [10, 12, 15, 19]. Inthis paper, the following theorem shall be proved. Theorem 1.1.
Let ( u, b ) and ( v, h ) be two weak solutions to system (1.1)-(1.3)such that for all t ≥ , | T | Z T u ( t, x )d x = 1 | T | Z T v ( t, x )d x = 0 and | T | Z T b ( t, x ) − h ( t, x )d x = 0 . ETERMINING WAVENUMBERS 3
Let Λ u,v ( t ) := max { Λ u ( t ) , Λ v ( t ) } and Λ b,h ( t ) := max { Λ b ( t ) , Λ h ( t ) } with Λ u ( t ) , Λ v ( t ) , Λ b ( t ) and Λ h ( t ) defined as in Definition 1.4-1.5. Let Q u,v ( t ) and Q b,h ( t ) besuch that Λ u,v ( t ) = λ Q u,v ( t ) and Λ b,h ( t ) = λ Q b,h ( t ) . If (cid:0) u ≤ Q u,v ( t ) ( t ) , b ≤ Q b,h ( t ) ( t ) (cid:1) = (cid:0) v ≤ Q u,v ( t ) ( t ) , h ≤ Q b,h ( t ) ( t ) (cid:1) , ∀ t > , then lim t →∞ (cid:0) k u ( t ) − v ( t ) k L + k b ( t ) − h ( t ) k L (cid:1) = 0 . Remark 1.2.
Due to the Galilean invariance of the fluid equation, it suffices toassume that u and v are zero-mean solutions. Yet, in general the magnetic fields donot have zero means, so in the above theorem it is assumed that the space-averageof b is the same as that of h. Preliminaries
Notation.
Throughout the paper, A . B denotes an estimate of the form A ≤ CB with some absolute constant C . Given a tempered distribution u, onedenotes by F u = ˆ u and F − u = ˇ u the Fourier transform and the inverse Fouriertransform of u, respectively. For simplicity, the L p -norm k·k L p is sometimes writtenas k · k p , while H s denotes the L -based Sobolev spaces.2.2. Well-posedness results for system (1.1)-(1.3).
In order to discuss thedetermining modes of the solutions, one had better first clarify the notions andexistence of the solutions. Relevant to this paper are the following well-posednessresults. From [1], it is known that in T , the Leray-Hopf type weak solution tosystem (1.1)-(1.3) exists globally in time, just as in the case of the NSE. Theorem 2.1 (Leray-Hopf type weak solution) . Let the initial data u , b ∈ L ( T ) . There exists a global weak solution ( u, b ) to system (1.1)-(1.3) satisfying ( u, b ) ∈ L ∞ (cid:0) , T ; ( L ( T )) (cid:1) ∩ L (cid:0) , T ; ( H ( T )) (cid:1) . In addition, the following energy inequality holds -
12 dd t (cid:0) k u k L + k b k L (cid:1) ≤ − ν k∇ u k L − µ k∇ b k L . In [5], local existence of strong solutions was proven. Furthermore, for smallinitial data, the existence of strong solutions is global.
Theorem 2.2 (Strong solution) . Let s > be an integer and u , b ∈ H s ( T ) with ∇ · u = ∇ · b = 0 . Then:i) The initial value ( u , b ) generates a local-in-time classical solution ( u, b ) ∈ L ∞ (cid:0) , T ; ( H s ( T ) ) (cid:1) with T = T ( k u k H s , k b k H s ) . ii) There exists a constant ǫ = ǫ ( ν, s ) such that ( u , b ) generates a global classicalsolution ( u, b ) ∈ L ∞ (cid:0) , ∞ ; ( H s ( T ) ) (cid:1) , provided that k u k H s + k b k H s < ǫ. To demonstrate the finite dimensional behaviour of the solutions, the followingregularity criterion, found in [6], is needed.
HAN LIU
Theorem 2.3 (Prodi-Serrin type regularity criterion) . Let s > be an integer and u , b ∈ H s ( T ) with ∇ · u = ∇ · b = 0 . Then for the first blow-up time T ∗ < ∞ of the classical solution to system (1.1)- (1.3), it holds that lim sup t ր T ∗ ( k u ( t ) k H s + k b ( t ) k H s ) = ∞ , if and only if k u k L q (0 ,T ∗ ; L p ( T )) + k∇ b k L γ (0 ,T ∗ ; L β ( T )) = ∞ , where p, q, β and γ satisfy the relation p + 2 q ≤ , β + 2 γ ≤ , with p, β ∈ (3 , ∞ ] . Littlewood-Paley decomposition.
This section is a brief introduction tothe Littlewood-Paley theory, a fundamental tool used throughout the paper. Onestarts with introducing a family of functions with annular support, { ϕ q ( ξ ) } ∞ q = − ,which forms a dyadic partition of unity in the frequency domain. Let λ q = 2 q , q ∈ Z . One chooses a radial function χ ∈ C ∞ ( R n ) satisfying χ ( ξ ) = ( , for | ξ | ≤ , for | ξ | ≥ , and define ϕ ( x ) = χ ( ξ ) − χ ( ξ ) and ϕ q ( ξ ) = ( ϕ ( λ − q ξ ) , for q ≥ ,χ ( ξ ) , for q = − . For a vector field u ∈ S ′ ( T n ) , one defines the Littlewood-Paley projections as∆ q u = u q =: X k ∈ Z n ϕ q ( k )ˆ u ( k ) e i πk · x , where ˆ u ( k ) is the k -th Fourier coefficient of u. In particular, ˆ u (0) = u − . Thus, atleast in the distributional sense u can be identified as a sum of its Littlewood-Paleyprojections u = ∞ X q = − u q . One also introduces the following notations, which appear throughout the paper, u ≤ Q := Q X q = − u q , u ( P,Q ] := Q X q = P +1 u q , ˜ u q := X | p − q |≤ u p . The L -based Sobolev spaces can thus be characterized via Littlewood-Paleyprojections - k u k H s = (cid:16) X q ≥− λ sq k u q k (cid:17) . In addition, the following Bernstein’s inequality shall be used extensively.
Lemma 2.4.
Let n be the space dimension and let s ≥ r ≥ , then k u q k r . λ n ( r − s ) q k u q k s . Proof:
See [3]. (cid:3)
ETERMINING WAVENUMBERS 5
Bony’s paraproduct and commutator estimates.
The product of twodistributions u and v can be formally written as uv = X p,q ≥− u p v q . Using Bony’s paradifferential calculus, one has the following paraproduct decom-position uv = X q ≥− u ≤ q − v q + X q ≥− u q v ≤ q − + X q ≥− ˜ u q v q , which distinguishes three parts in the product uv. To facilitate the estimations, one introduces the following commutators for theconvection or inertial terms and the Hall term, respectively(2.6) [∆ q , u ≤ p − · ∇ ] v p = ∆ q ( u ≤ p − · ∇ v p ) − u ≤ p − · ∇ ∆ q v p , (2.7) [∆ q , b ≤ p − × ∇× ] h p = ∆ q ( b ≤ p − × ( ∇ × h p )) − b ≤ p − × ( ∇ × ∆ q h p ) . In the upcoming sections, it shall be seen that the commutators, along withthe divergence free conditions, reveal certain cancellations within the nonlinearinteractions. The commutator (2.6) enjoys the following estimate, proven in [3].
Lemma 2.5.
For r = r + r , the following inequality is true - k [∆ q , u ≤ p − · ∇ ] v p k r . k v p k r X p ′ ≤ p − λ p ′ k u p ′ k r . The commutator (2.7) satisfies an analogous estimate, as shown in [15].
Lemma 2.6.
Given that ∇ · b ≤ p − = 0 , the following inequality holds - k [∆ q , b ≤ p − × ∇× ] h p k r . k h p k r X p ′ ≤ p − λ p ′ k b p ′ k ∞ . More detailed study of the aforementioned harmonic analysis tools and theirapplications can be found in the work of Bahouri, Chemin and Danchin [3].3.
Analysis of a reduced system
To analyze the complete Hall-MHD system, one starts by considering the fluid-free version of system (1.1)-(1.3), written as follows - b t + η ∇ × (( ∇ × b ) × b ) = µ ∆ b, (3.8) ∇ · b = 0 . (3.9)The above system is named electron magneto-hydrodynamic (EMHD) equationsas it describes the situation where the ions in the Hall-MHD setting are too heavyto move, leaving only the electrons in motion. As the small-scale limit of theHall-MHD system, the EMHD equations can be used as a toy model to betterunderstand the Hall term. In [1], the existence of weak solutions to (3.8)-(3.9) onperiodic domains, analogous to that of the complete Hall-MHD system, was shown.For more studies concerning the EMHD equations, readers may consult [30, 33, 41].In the following passages, b and h are two weak solutions to system (3.8)-(3.9).One aims to prove the following analogue of Theorem 1.1. HAN LIU
Theorem 3.1.
Let Λ b,h ( t ) := max { Λ b ( t ) , Λ h ( t ) } with Λ b ( t ) and Λ h ( t ) defined asin definition (1.5). Let Q b,h ( t ) be such that Λ b,h ( t ) = λ Q b,h ( t ) . If b ≤ Q b,h ( t ) ( t ) = h ≤ Q b,h ( t ) ( t ) , ∀ t > , then lim t →∞ k b ( t ) − h ( t ) k L = 0 , ∀ s > . Proof:
Straightforward calculations show that m := b − h satisfies m t − µ ∆ m = − η ∇ × (( ∇ × m ) × h ) − η ∇ × (( ∇ × b ) × m ) . Multiplying the above equation by λ sq ∆ q m, integrating by parts and summingover q lead to the following identity.12 dd t X q ≥− λ sq k m q k + µ X q ≥− λ s +2 q k m q k = η X q ≥− λ sq Z T ∆ q (( ∇ × m ) × h ) · ( ∇ × m q )d x + η X q ≥− λ sq Z T ∆ q (( ∇ × b ) × m ) · ( ∇ × m q )d x =: I + J. One further decomposes the terms I and J using Bony’s paraproduct. I = η X q ≥− X | p − q |≤ λ sq Z T ∆ q (( ∇ × m p ) × h ≤ p − ) · ( ∇ × m q )d x + η X q ≥− X | p − q |≤ λ sq Z T ∆ q (( ∇ × m ≤ p − ) × h p ) · ( ∇ × m q )d x + η X q ≥− X p ≥ q − λ sq Z T ∆ q (( ∇ × ˜ m p ) × h p ) · ( ∇ × m q )d x =: I + I + I ; J = η X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m ≤ p − × ( ∇ × b p )) · ( ∇ × m q )d x + η X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m p × ( ∇ × b ≤ p − )) · ( ∇ × m q )d x + η X q ≥− X p ≥ q − λ sq Z T ∆ q ( ˜ m p × ( ∇ × b p )) · ( ∇ × m q )d x =: J + J + J . One then proceeds to estimate the terms I , I , I and J , J , J . As for I , onerewrites it using the commutator notation (2.6) and notices that I in the following ETERMINING WAVENUMBERS 7 expression vanishes. I = η X q ≥− X | p − q |≤ λ sq Z T (cid:0) [∆ q , h ≤ p − × ∇× ] m p (cid:1) · ( ∇ × m q )d x − η X q ≥− λ sq Z T (cid:0) h ≤ q − × ( ∇ × m q ) (cid:1) · ( ∇ × m q )d x + η X q ≥− X | p − q |≤ λ sq Z T (cid:0) ( h ≤ q − − h ≤ p − ) × ( ∇ × ( m p ) q ) (cid:1) · ( ∇ × m q )d x =: I + I + I . Taking into account that m ≤ Q b,h = 0 , one splits I by the wavenumber. I = η X q>Q b,h X | p − q |≤ λ sq Z T (cid:0) [∆ q , h ≤ Q b,h × ∇× ] m p (cid:1) · ( ∇ × m q )d x + η X q>Q b,h X | p − q |≤ λ sq Z T (cid:0) [∆ q , h ( Q h ,p − × ∇× ] m p (cid:1) · ( ∇ × m q )d x =: I + I . By Lemma 2.5, H¨older’s inequality, Definition 1.5, Young’s inequalities, oneestimates I as follows. | I | ≤ η k∇ h ≤ Q b,h k ∞ X q>Q h λ sq k m q k X | p − q |≤ k∇ × m p k ≤ η k h ≤ Q b,h k ∞ X q>Q h λ s +1 q k m q k X | p − q |≤ k∇ × m p k . c r µ X q ≥− λ sq k∇ m q k . One estimates I using Lemma 2.5, H¨older’s inequality, Definition 1.5, Young’sand Jensen’s inequalities. | I | ≤ η X q>Q h λ sq k∇ × m q k X | p − q |≤ k m p k X Q h
Q h λ sq k∇ × m q k X | p − q |≤ k∇ × m p k X Q h
Q h λ sq k∇ × m q k X | p − q |≤ k∇ × m p k X Q h
Q b,h X | p − q |≤ λ sq Z T | h q − ||∇ × ( m p ) q ||∇ × m q | d x. HAN LIU
The sum is then split by the wavenumber Q b,h . | I | . η X Q b,h Q b,h +2 X | p − q |≤ λ sq Z T | h q − ||∇ × ( m p ) q ||∇ × m q | d x =: I + I .I is estimated as follows. I ≤ η k h ≤ Q b,h k ∞ X Q b,h
Q b,h +2 λ sq k h q − k ∞ k∇ × m q k X | p − q |≤ k∇ × m p k ≤ c r µ X q>Q b,h +2 λ sq k∇ × m q k X | p − q |≤ k∇ × m p k . c r µ X q ≥− λ sq k∇ m q k . As m ≤ Q b,h = 0 , it is perceivable that I consists of only high frequency partsand can be written as follows. I = η X p>Q b,h +2 X | p − q |≤ λ sq Z T ∆ q (cid:0) h p × ( ∇ × m ( Q b,h ,p − ) (cid:1) · ( ∇ × m q )d x. Let δ > s.
H¨older’s, inequality, Definition 1.5, Young’s and Jensen’s inequalitieslead to | I | ≤ η X p>Q b,h k h p k ∞ X | p − q |≤ λ sq k∇ × m q k X Q h
Q b,h − λ sq k∇ × m q k X | p − q |≤ X Q b,h
Q b,h − λ sq k∇ × m q k X | p − q |≤ X Q b,h
Q b,h − λ sq k∇ × m q k X Q b,h
Q b,h λ sq Z T ∆ q ( h p × ( ∇ × ˜ m p )) · ∇ × m q d x + η X q>Q b,h +2 X p ≥ q − λ sq Z T ∆ q ( h p × ( ∇ × ˜ m p )) · ∇ × m q d x =: I + I + I . Invoking Definition 1.5 and applying H¨older’s, Young’s and Jensen’s inequalities,one can estimate I , I and I as follows. | I | ≤ η X Q b,h Q b,h λ sq k∇ × m q k X | p − q |≤ X Q b,h Q b,h λ sq k∇ × m q k X Q b,h Q b,h X | p − q |≤ λ sq Z T ∆ q (cid:0) m p × ( ∇ × b ≤ Q b,h ) (cid:1) · ( ∇ × m q )d x + X q>Q b,h X | p − q |≤ λ sq Z T ∆ q (cid:0) m p × ( ∇ × b ( Q b,h ,p − ) (cid:1) · ( ∇ × m q )d x =: J + J . To estimate J , one applies H¨older’s and Young’s inequalities. | J | ≤ η k∇ b ≤ Q b,h k ∞ X q>Q b,h λ sq k∇ × m q k X | p − q |≤ k m p k . η k b ≤ Q b,h k ∞ X q>Q b,h λ s +1 q k∇ m q k X | p − q |≤ k m p k . c r µ X q>Q b,h λ sq k∇ m q k X | p − q |≤ λ s +1 p k m p k . c r µ X q ≥− λ sq k∇ m q k . For J , H¨older’s inequality, Definition 1.5, Young’s and Jensen’s inequalities yield | J | ≤ η X q>Q b,h λ sq k∇ × m q k X | p − q |≤ k m p k X Q b,h Q b,h λ sq k∇ × m q k X | p − q |≤ λ s +1 p k m p k X Q b,h Q b,h X p ≥ q +2 λ sq Z T ∆ q (cid:0) ˜ m p × ( ∇ × b p ) (cid:1) · ( ∇ × m q )d x, ETERMINING WAVENUMBERS 11 which can then be estimated as follows. | J | ≤ η X p ≥ Q b,h λ p k b p k ∞ k m p k X q ≤ p − λ sq k∇ × m q k ≤ c r µ X q>Q b,h − λ sq k∇ × m q k X p ≥ q +2 λ p k m p k ≤ c r µ X q>Q b,h − λ sq k∇ × m q k X p ≥ q +2 λ s +1 p k m p k λ sq − p λ δQ b,h − p . c r µ X q ≥− λ sq k∇ m q k . Let c r = 1 − (2 µ ) − . Assembling all the estimates above leads todd t X q ≥− λ sq k m q k . − X q ≥− λ s +2 q k m q k . − λ X q ≥− λ sq k m q k . Setting s = 0 , one sees that dd t k m k L . − λ k m k L . The desired result then followsfrom Gr¨onwall’s inequality. (cid:3) Proof of Theorem 1.1 Let ( u, b ) and ( v, h ) be two weak solutions to system (1.1)-(1.3). Straightforwardcalculations show that the difference ( w, m ) := ( u − v, b − h ) satisfies the followingsystem of equations. w t − ν ∆ w = − u · ∇ w − w · ∇ v + b · ∇ m + m · ∇ h − ∇ π,m t − µ ∆ m = − v · ∇ m − w · ∇ b + b · ∇ w + m · ∇ v − ∇ × ( ∇ × m ) × h ) − ∇ × (( ∇ × b ) × m ) . (4.10)Utilizing the wavenumbers, one shall eventually prove that ( w, m ) satisfies thefollowing inequality(4.11) dd t (cid:0) k w k L + k m k L (cid:1) . − (cid:0) k∇ w k L + k∇ m k L (cid:1) , which leads to theorem (1.1).To this end, one considers a frequency-localized version of system (4.10) in energyspaces. Multiplying the equations by λ sq ∆ q w and λ sq ∆ q m respectively, integratingby parts and summing over q, one obtains12 dd t X q ≥− λ sq k w q k + ν X q ≥− λ s +2 q k w q k ≤ − X q ≥− λ sq Z T ∆ q ( u · ∇ w ) · w q d x − X q ≥− λ sq Z T ∆ q ( w · ∇ v ) · w q d x + X q ≥− λ sq Z T ∆ q ( b · ∇ m ) · w q d x + X q ≥− λ sq Z T ∆ q ( m · ∇ h ) · w q d x =: A + B + C + D, (4.12) and 12 dd t X q ≥− λ sq k m q k + µ X q ≥− λ s +2 q k m q k ≤ − X q ≥− λ sq Z T ∆ q ( v · ∇ m ) · m q d x − λ sq Z T ∆ q ( w · ∇ b ) · m q d x + X q ≥− λ sq Z T ∆ q ( b · ∇ w ) · m q d x + X q ≥− λ sq Z T ∆ q ( m · ∇ v ) · m q d x − X q ≥− λ sq Z T ∆ q (( ∇ × m ) × h ) · ( ∇ × m q )d x − X q ≥− λ sq Z T ∆ q (( ∇ × b ) × m ) · ( ∇ × m q )d x =: E + F + G + H + I + J. (4.13)The tasks are then to control the terms A – J. Estimation of A. The estimates for A fall into the same line as those in [11].Bony’s decomposition leads to the following - A = − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( u ≤ p − · ∇ w p ) · w q d x − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( u p · ∇ w ≤ p − ) · w q d x − X q ≥− X p ≥ q − λ sq Z T ∆ q ( u p · ∇ ˜ w p ) · w q d x =: A + A + A . Using Definition 1.4, one then separate the lower and higher modes of A . | A | ≤ X p>Q u,v X | q − p |≤ λ sq Z T | ∆ q ( u ≤ p − · ∇ w p ) · w q | d x ≤ X p>Q u,v X | q − p |≤ λ sq Z T | ∆ q ( u ≤ Q u,v · ∇ w p ) · w q | d x + X p ′ >Q u,v X p ≥ p ′ +2 X | q − p |≤ λ sq Z T | ∆ q ( u p ′ · ∇ w p ) · w q | d x =: A + A . ETERMINING WAVENUMBERS 13 To control the lower modes, one uses Definition 1.4, Lemma 2.4, H¨older’s andYoung’s inequalities. A . k u ≤ Q u,v k r X p>Q u,v λ p k w p k X | q − p |≤ λ sq k w q k rr − . c r ν X p>Q u,v λ p k w p k X | q − p |≤ λ − r Q u,v λ s + r q k w q k . c r ν X q ≥− λ sq k∇ w q k . The higher modes are estimated as follows. A . X p ′ >Q u,v k u p ′ k r X p>p ′ +2 λ p k w p k X | q − p |≤ λ sq k w q k rr − . c r ν X p ′ >Q u,v λ − r p ′ X p>p ′ +2 λ p k w p k X | q − p |≤ λ s + r q k w q k . c r ν X q ≥− λ sq k∇ w q k . It follows from the condition w ≤ Q u,v = 0 that A = − X p>Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( u p · ∇ w ≤ p − ) · w q d x. Recalling Definition 1.4, one then estimates A using H¨older’s and Young’s in-equalities. | A | ≤ X p>Q u,v +2 k u p k r X Q u,v Q u,v +2 λ − r p X Q u,v Q u,v λ s +1 q k w q k X Q u,v Q u,v X q ≤ p +2 λ sq Z T ∆ q ( u p · ∇ ˜ w p ) · w q d x =: A + A . One has no difficulty in controlling the few lower modes. | A | . Λ u,v k u Q u,v k r k w Q u,v k X Q u,v As a result of Bony’s paraproduct decomposition B = − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( w ≤ p − · ∇ v p ) · w q d x − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( w p · ∇ v ≤ p − ) · w q d x − X q ≥− X p ≥ q − λ sq Z T ∆ q ( w p · ∇ ˜ v p ) · w q d x =: B + B + B . Since w ≤ Q u,v = 0 , B consists of only higher modes. B = − X p>Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( w ≤ p − · ∇ v p ) · w q d x. ETERMINING WAVENUMBERS 15 Let 1 − r < . One can estimate B using Definition 1.4, H¨older’s, Young’s andJensen’s inequalities. | B | . X p>Q u,v +2 λ p k v p k r X Q u,v Q u,v +2 λ − r p X Q u,v Q u,v +2 λ s +2 − r p k w p k X Q u,v Q u,v +2 λ s +1 p k w p k X Q u,v Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( w p · ∇ v ≤ Q u,v ) · w q d x − X p>Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( w p · ∇ v ( Q u,v ,p − ) · w q d x =: B + B + B . The estimate for the lower modes | B | + | B | are as follows. | B | + | B | . X p>Q u,v k w p k rr − X | q − p |≤ λ sq k w q k X p ′ Q u,v +2 k w p k X Q u,v Q u,v +2 λ s +1 p k w p k X | q − p |≤ λ s +1 q k w q k X Q u,v Q u,v +1 X q ≤ p +2 λ sq Z T | ∆ q ( w p · ∇ v p − ) · w q | d x =: B + B . The term B , consisting of scarce lower modes, can be controlled with ease. B . Λ s +1 u,v k w Q u,v k k v Q u,v k r X Q u,v Bony’s paraproduct decomposition yields C = X q ≥− X | p − q |≤ λ sq Z T ∆ q ( b ≤ p − · ∇ m p ) w q d x + X q ≥− X | p − q |≤ λ sq Z T ∆ q ( b p · ∇ m ≤ p − ) w q d x + X q ≥− X p ≥ q − λ sq Z T ∆ q ( b p · ∇ ˜ m p ) w q d x =: C + C + C . ETERMINING WAVENUMBERS 17 Moreover, one rewrites C using the commutator as C = X q ≥− X | p − q |≤ λ sq Z T [∆ q , b ≤ p − · ∇ ] m p w q d x + X q ≥− X | p − q |≤ λ sq Z T b ≤ q − · ∇ ∆ q m p w q d x + X q ≥− X | p − q |≤ λ sq Z T ( b ≤ p − − b ≤ q − ) · ∇ ∆ q m p w q d x =: C + C + C . As will be seen later, C cancels a part of the term G. Taking into account that m ≤ Q b,h = 0 , one split C using the wavenumber Q b,h .C = X Q b,h Q b,h +2 X | q − p |≤ λ sq Z T [∆ q , b ≤ Q b,h · ∇ ] m p w q d x + X p>Q b,h +2 X | q − p |≤ λ sq Z T [∆ q , b ( Q b,h ,p − · ∇ ] m p w q d x =: C + C + C . By Definition 1.5, H¨older’s and Young’s inequalities, the following estimate holds. | C | + | C | ≤k∇ b ≤ Q b,h k ∞ X p>Q b,h k m p k X | q − p |≤ λ sq k w q k ≤ c r κ X p ≥− λ s +1 p k m p k X | q − p |≤ λ s +1 q k w q k ≤ c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . As a result of Definition 1.5, H¨older’s and Young’s inequalities, the followingestimate for C is true. | C | ≤ X p>Q b,h k m p k X | q − p |≤ λ sq k w q k X Q b,h Q b,h k m p k X | q − p |≤ λ s +2 q k w q k X Q b,h Q b,h λ s +1 p k m p k X | q − p |≤ λ s +1 q k w q k X Q b,h Q b,h X | p − q |≤ λ sq Z T | b q ||∇ ∆ q m p w q | d x =: C + C . The estimate for C is as follows. C ≤ X − ≤ q ≤ Q b,h k b q k ∞ λ sq k w q k X | p − q |≤ k∇ m p k . c r κ X q ≥− λ s +1 q k w q k X | p − q |≤ λ s +1 p k m p k . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) C enjoys the following estimate, thanks to Definition 1.5. C ≤ X q>Q b,h k b q k ∞ λ sq k w q k X | p − q |≤ k∇ m p k . c r κ X q ≥− λ s +1 q k w q k X | p − q |≤ λ s +1 p k m p k . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k )Since ∇ m ≤ Q b,h = 0 , the lower modes of C vanish and it can be seen that C = X p>Q b,h +2 X | p − q |≤ λ sq Z T ∆ q ( b p · ∇ m ( Q b,h ,p − ) w q d x, which is estimated using H¨older’s, Young’s and Jensen’s inequalities as | C | ≤ X p>Q b,h +2 k b p k ∞ X | p − q |≤ λ sq k w q k X Q b,h Q b,h +2 k b p k ∞ X | p − q |≤ λ s +1 q k w q k X Q b,h Q b,h +2 X | p − q |≤ λ s +1 q k w q k X Q b,h
One splits C into lower and higher modes. C = X Q b,h − ≤ p ≤ Q b,h X q ≤ p +2 λ sq Z T ∆ q ( b p · ∇ ˜ m p ) w q d x + X p>Q b,h X q ≤ p +2 λ sq Z T ∆ q ( b p · ∇ ˜ m p ) w q d x =: C + C .C , made up from the scarce lower modes, is estimated as follows. | C | ≤ X Q b,h − ≤ p ≤ Q b,h k b p k ∞ k∇ ˜ m p k X q ≤ p +2 λ sq k w q k ≤ c r κ X Q b,h − ≤ p ≤ Q b,h λ s +1 p k m p k X q ≤ p +2 λ s +1 q k w q k λ sq − p ≤ c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . One recalls Definition 1.5 and applies H¨older’s, Young’s and Jensen’s inequalitiesto bound C . | C | ≤ X p>Q b,h k b p k ∞ k∇ ˜ m p k X q ≤ p +2 λ sq k w q k ≤ c r κ X p>Q b,h λ s +1 p k m p k X q ≤ p +2 λ s +1 q k w q k λ sq − p ≤ c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . Estimation of D. Bony’s paraproduct decomposition yields D = X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m p · ∇ h ≤ p − ) w q d x + X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m ≤ p − · ∇ h p ) w q d x + X q ≥− X p ≥ q − λ sq Z T ∆ q ( ˜ m p · ∇ h p ) w q d x =: D + D + D . Utilizing the wavenumber Q b,h , one splits D into three terms. D = X Q b,h Q b,h +2 X | q − p |≤ λ sq Z T ∆ q ( m p · ∇ h ≤ Q b,h ) w q d x + X p>Q b,h +2 X | q − p |≤ λ sq Z T ∆ q ( m p · ∇ h ( Q b,h ,p − ) w q d x =: D + D + D 130 HAN LIU One can estimate | D | + | D | without difficulties. | D | + | D | ≤k∇ h ≤ Q b,h k ∞ X p>Q b,h k m p k X | q − p |≤ λ sq k w q k ≤ c r κ X p>Q b,h λ p k m p k X | q − p |≤ λ sq k w q k ≤ c r κ X p>Q b,h λ s +1 p k m p k X | q − p |≤ λ s +1 q k w q k ≤ c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . By Definition 1.5, H¨older’s, Young’s and Jensen’s inequalities, one has | D | ≤ X p>Q b,h +2 k m p k X | q − p |≤ λ sq k w q k X Q b,h Q b,h +2 λ p k m p k X | q − p |≤ λ sq k w q k X Q b,h Q b,h +2 λ s +1 p k m p k X | q − p |≤ λ s +1 q k w q k X Q b,h Q b,h +2 X | q − p |≤ λ sq Z T ∆ q ( m ( Q b,h ,p − · ∇ h p ) w q d x. Using H¨older’s Young’s and Jensen’s inequalities, one estimates D . | D | ≤ X p>Q b,h +2 λ p k h p k ∞ X | q − p |≤ λ sq k w q k X Q b,h Q b,h λ s +1 q k w q k X Q b,h Q b,h λ s +1 q k w q k X Q b,h Q b,h X q ≤ p +2 λ sq Z T ∆ q ( ˜ m p · ∇ h p ) w q d x =: D + D . ETERMINING WAVENUMBERS 21 D satisfies the following estimate. D ≤k∇ h Q b,h k ∞ X − ≤ q ≤ Q b,h +2 λ sq k w q k k m Q b,h +1 k . c r κλ s +1 Q b,h +1 k m Q b,h +1 k X − ≤ q ≤ Q b,h +2 λ s +1 q k w q k . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . The estimate for D follows from Definition 1.5, H¨older’s, Young’s and Jensen’sinequalities. D ≤ X p>Q b,h k∇ h p k ∞ k ˜ m p k X q ≤ p +2 λ sq k w q k . c r κ X p>Q b,h λ p k ˜ m p k X q ≤ p +2 λ sq k w q k . c r κ X p>Q b,h λ s +1 p k m p k X q ≤ p +2 λ s +1 q k w q k λ s − q − p . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . Estimation of E. One decomposes E using Bony’s paraproduct. E = − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( v p · ∇ m ≤ p − ) m q d x − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( v ≤ p − · ∇ m p ) m q d x − X q ≥− X p ≥ q − λ sq Z T ∆ q ( v p · ∇ ˜ m p ) m q d x =: E + E + E . Utilizing the wavenumber Q u,v , E is split into two. E = − X p ≤ Q u,v X | q − p |≤ λ sq Z T ∆ q ( v p · ∇ m ≤ p − ) m q d x − X p>Q u,v X | q − p |≤ λ sq Z T ∆ q ( v p · ∇ m ≤ p − ) m q d x =: E + E . By Definition 1.4, H¨older’s, Young’s and Jensen’s inequalities, E and E areestimated in the following ways. | E | ≤ X p ≤ Q u,v k v p k r k∇ m ≤ p − k rr − X | q − p |≤ λ sq k m q k ≤ X p ≤ Q u,v λ − r p k v p k r X p ′ ≤ p − λ r p ′ k m p ′ k X | q − p |≤ λ s +1 − r q k m q k . c r κ X q ≤ Q u,v +2 λ s +1 q k m q k X p ′ ≤ q λ s +1 p ′ k m p ′ k λ s − r q − p ′ . c r κ X q ≥− λ s +2 q k m q k ; | E | ≤ X p>Q u,v k v p k r k∇ m ≤ p − k rr − X | q − p |≤ λ sq k m q k ≤ X p>Q u,v λ − r p k v p k r X p ′ ≤ p − λ r p ′ k m p ′ k X | q − p |≤ λ s +1 − r q k m q k . c r κ X q>Q u,v − λ s +1 q k m q k X p ′ ≤ q λ s +1 p ′ k m p ′ k λ s − r q − p ′ . c r κ X q ≥− λ s +2 q k m q k . By the commutator notation, one has E = X q ≥− X | p − q |≤ λ sq Z T [∆ q , v ≤ p − · ∇ ] m p m q d x + X q ≥− X | p − q |≤ λ sq Z T v ≤ q − · ∇ ∆ q m p m q d x + X q ≥− X | p − q |≤ λ sq Z T ( v ≤ p − − v ≤ q − ) · ∇ ∆ q m p m q d x =: E + E + E , where E vanishes as ∇ · v ≤ q − = 0 . Splitting E by the wavenumber Q u,v , one has E = X − ≤ p ≤ Q u,v +2 X | q − p |≤ λ sq Z T [∆ q , v ≤ p − · ∇ ] m p m q d x + X p>Q u,v +2 X | q − p |≤ λ sq Z T [∆ q , v ≤ Q u,v · ∇ ] m p m q d x + X p>Q u,v +2 X | q − p |≤ λ sq Z T [∆ q , v ( Q u,v ,p − · ∇ ] m p m q d x =: E + E + E . ETERMINING WAVENUMBERS 23 Using Definition 1.4, Lemma 2.5, H¨older’s and Young’s inequalities, one canestimate E . | E | ≤ X − ≤ p ≤ Q u,v +2 k m p k rr − X | q − p |≤ λ sq k m q k X p ′ ≤ p − λ p ′ k v p ′ k r . c r κ X − ≤ p ≤ Q u,v +2 λ r p k m p k X | q − p |≤ λ sq k m q k λ − r p . c r κ X − ≤ p ≤ Q u,v +2 λ s +1 p k m p k X | q − p |≤ λ s +1 q k m q k . c r κ X q ≥− λ s +2 q k m q k . The term E can be estimated in a similar fashion. | E | ≤ X p>Q u,v +2 k m p k rr − X | q − p |≤ λ sq k m q k X p ′ ≤ Q u,v λ p ′ k v p ′ k r . c r κ X p>Q u,v +2 λ r p k m p k X | q − p |≤ λ sq k m q k λ − r p . c r κ X p>Q u,v +2 λ s +1 p k m p k X | q − p |≤ λ s +1 q k m q k . c r κ X q ≥− λ s +2 q k m q k . The estimate for E follows from Definition 1.4, Lemma 2.5, H¨older’s andYoung’s inequalities. | E | ≤ X p>Q u,v +2 k m p k rr − X | q − p |≤ λ sq k m q k X Q u,v Q u,v +2 λ r p k m p k X | q − p |≤ λ sq k m q k X Q u,v Q u,v +2 λ s +1 p k m p k X | q − p |≤ λ s +1 q k m q k X Q u,v Q u,v X | p − q |≤ λ sq Z T | v q ||∇ ∆ q m p m q | d x =: E + E . The estimate for E is as follows. E . X − ≤ q ≤ Q u,v λ sq k v q k r k m q k rr − X | p − q |≤ k∇ m p k . c r κ X − ≤ q ≤ Q u,v λ s +1 q k m q k X | p − q |≤ λ p k m p k . c r κ X − ≤ q ≤ Q u,v λ s +1 q k m q k X | p − q |≤ λ s +1 p k m p k . c r κ X q ≥− λ s +2 q k m q k . By Definition 1.4, H¨older’s and Young’s inequalities, E can be estimated. E ≤ X q>Q u,v λ sq k v q k r k m q k rr − X | p − q |≤ k∇ m p k . c r κ X q>Q u,v λ s +1 q k m q k X | p − q |≤ λ p k m p k . c r κ X q>Q u,v λ s +1 q k m q k X | p − q |≤ λ s +1 p k m p k . c r κ X q ≥− λ s +2 q k m q k . One separates the lower and higher modes of E using the wavenumber Q u,v .E = − X − ≤ p ≤ Q u,v X q ≤ p − λ sq Z T ∆ q ( v p · ∇ ˜ m p ) m q d x − X p>Q u,v X q ≤ p − λ sq Z T ∆ q ( v p · ∇ ˜ m p ) m q d x =: E + E . With the help of Definition 1.4, H¨older’s, Young’s and Jensen’s inequalities, theterms E and E can be under control. | E | ≤ X − ≤ p ≤ Q u,v k v p k r k∇ m p k X q ≤ p − λ sq k m q k rr − ≤ c r κ X − ≤ p ≤ Q u,v λ − r p k m p k X q ≤ p − λ sq k m q k rr − ≤ c r κ X − ≤ p ≤ Q u,v λ − r p k m p k X q ≤ p − λ s + r q k m q k ≤ c r κ X − ≤ p ≤ Q u,v λ s +1 p k m p k X q ≤ p − λ s +1 q k m q k λ s + r − q − p ≤ c r κ X q ≥− λ s +2 q k m q k ; ETERMINING WAVENUMBERS 25 | E | ≤ X p>Q u,v k v p k r k∇ m p k X q ≤ p − λ sq k m q k rr − ≤ c r κ X p>Q u,v λ − r p k m p k X q ≤ p − λ sq k m q k rr − ≤ c r κ X p>Q u,v λ − r p k m p k X q ≤ p − λ s + r q k m q k ≤ c r κ X p>Q u,v λ s +1 p k m p k X q ≤ p − λ s +1 q k m q k λ s + r − q − p ≤ c r κ X q ≥− λ s +2 q k m q k . Estimation of F. By Bony’s paraproduct decomposition, one has F = − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( w p · ∇ b ≤ p − ) m q d x − X q ≥− X | p − q |≤ λ sq Z T ∆ q ( w ≤ p − · ∇ b p ) m q d x − X q ≥− X p ≥ q − λ sq Z T ∆ q ( ˜ w p · ∇ b p ) m q d x =: F + F + F . Using the fact that m ≤ Q b,h = 0 , one splits F into two terms. F = − X p>Q b,h +2 X | q − p |≤ λ sq Z T ∆ q ( w p · ∇ b ≤ Q b,h ) m q d x − X p>Q b,h +2 X | q − p |≤ λ sq Z T ∆ q ( w p · ∇ b ( Q b,h ,p − ) m q d x =: F + F . To estimate F , one uses Definition 1.5, H¨older’s and Young’s inequalities. | F | ≤k∇ b ≤ Q b,h k ∞ X q>Q b,h λ sq k m q k X | p − q |≤ k w p k ≤k b ≤ Q b,h k ∞ X q>Q b,h λ s +1 q k m q k X | p − q |≤ λ sp k w p k ≤ c r κ X q>Q b,h λ s +1 q k m q k X | p − q |≤ λ s +1 p k w p k ≤ c r κ X q> − ( λ s +2 q k w q k + λ s +2 q k m q k ) . By Definition 1.5, H¨older’s and Young’s inequalities, F satisfies the following. | F | ≤ X q>Q b,h λ sq k m q k X | p − q |≤ k w p k X Q b,h Q b,h λ s +1 q k m q k X | p − q |≤ λ p k w p k X Q b,h Q b,h λ s +1 q k m q k X | p − q |≤ λ s +1 p k w p k X Q b,h − ( λ s +2 q k w q k + λ s +2 q k m q k ) .F is split into lower and higher modes based on the wavenumber Q b,h as wellas the fact that m ≤ Q b,h = 0 .F = − X Q b,h − Q b,h X | q − p |≤ λ sq Z T ∆ q ( w ≤ p − · ∇ b p ) m q d x =: F + F . It follows from Definition 1.5, H¨older’s, Young’s and Jensen’s inequalities that | F | ≤ X Q b,h − − ( λ s +2 q k w q k + λ s +2 q k m q k ) . One estimates F with the help of H¨older’s, Young’s and Jensen’s inequalities. | F | ≤ X p>Q b,h k∇ b p k ∞ X | q − p |≤ λ sq k m q k X p ′ ≤ p − k w p ′ k ≤ X q>Q b,h λ sq k m q k λ − δp Λ δb,h X | p − q |≤ λ δp − Q b,h k b p k ∞ X p ′ ≤ p − λ p ′ k w p ′ k λ − p ′ ≤ c r κ X q>Q b,h λ s +1 q k m q k X − ≤ p ′ ≤ q λ s +1 p ′ k w p ′ k λ sq − p ′ λ − p ′ ≤ c r κ X q> − ( λ sq k w q k + λ s +2 q k m q k ) . ETERMINING WAVENUMBERS 27 As m ≤ Q b,h = 0 , one splits F into two terms. F = − X p ≤ Q b,h X Q b,h Using Bony’s paraproduct decomposition, one has G = X q ≥− X | p − q |≤ λ sq Z T ∆ q ( b p · ∇ w ≤ p − ) m q d x + X q ≥− X | p − q |≤ λ sq Z T ∆ q ( b ≤ p − · ∇ w p ) m q d x + X q ≥− X p ≥ q − λ sq Z T ∆ q ( b p · ∇ ˜ w p ) m q d x =: G + G + G . Taking into account that m ≤ Q b,h = 0 , one separates lower and higher modes of G by the wavenumber Q b,h .G = X Q b,h − ≤ p ≤ Q b,h X | q − p |≤ λ sq Z T ∆ q ( b p · ∇ w ≤ p − ) m q d x + X p>Q b,h X | q − p |≤ λ sq Z T ∆ q ( b p · ∇ w ≤ p − ) m q d x =: G + G . Thanks to the fact that q = Q b,h + 1 or Q b,h + 2, one can control G . | G | ≤ X Q b,h − Q b,h k b p k ∞ X − ≤ p ′ ≤ p − λ p ′ k w p ′ k X | q − p |≤ λ sq k m q k ≤ c r κ X p>Q b,h X − ≤ p ′ ≤ p − λ s +1 p ′ k w p ′ k λ − sp ′ X | q − p |≤ λ s +1 q k m q k λ s − q . c r κ X p>Q b,h λ s +1 p k m p k X − ≤ p ′ ≤ p − λ s +1 p ′ k w p ′ k λ − sp ′ λ s − p . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . Rewriting G using the commutator notation yields G = X q ≥− X | p − q |≤ λ sq Z T [∆ q , b ≤ p − · ∇ ] w p m q d x + X q ≥− X | p − q |≤ λ sq Z T b ≤ q − · ∇ ∆ q w p m q d x + X q ≥− X | p − q |≤ λ sq Z T ( b p − − b q − ) · ∇ ∆ q w p m q d x =: G + G + G . ETERMINING WAVENUMBERS 29 One further splits G into three parts by the wavenumber Q b,h .G = X Q b,h − Q b,h +2 X | q − p |≤ λ sq Z T [∆ q , b ≤ Q b,h · ∇ ] w p m q d x + X p>Q b,h +2 X | q − p |≤ λ sq Z T [∆ q , b ( Q b,h ,p − · ∇ ] w p m q d x =: G + G + G . Using Definition 1.5, H¨older’s and Young’s inequalities, one can estimate | G | + | G | . | G | + | G | ≤k∇ b ≤ Q b,h k ∞ X p>Q b,h − k w p k X | q − p |≤ λ sq k m q k . k b ≤ Q b,h k ∞ X p>Q b,h − λ p k w p k X | q − p |≤ λ sq k m q k . c r κ X p ≥− λ s +1 p k w p k X | q − p |≤ λ s +1 q k m q k . c r κ X p ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . The estimate for G is as follows. | G | ≤ X p>Q b,h +2 k w p k X Q h,b Q b,h X | q − p |≤ λ sq Z T | b p ||∇ ∆ q w p m q | d x =: G + G . By Definition 1.5, H¨older’s and Young’s inequalities, one has G . X − ≤ p>Q b,h k b p k ∞ λ p k w p k X | q − p |≤ λ sq k m q k . c r κ X p ≥− λ s +1 p k w p k X | q − p |≤ λ s +1 q k m q k . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) .G is estimated as follows. G . X p>Q b,h k b p k ∞ λ p k w p k X | q − p |≤ λ sq k m q k . c r κ X p ≥− λ s +1 p k w p k X | q − p |≤ λ s +1 q k m q k . c r κ X q> − ( λ s +2 q k w q k + λ s +2 q k m q k ) . One divides G into lower and higher modes using the wavenumber Q b,h .G = X Q b,h − Q b,h X q ≤ p +2 λ sq Z T ∆ q ( b p · ∇ ˜ w p ) m q d x =: G + G . One can estimate G in the following way. | G | ≤ X Q b,h −
Meanwhile, by Definition 1.5, H¨older’s, Young’s and Jensen’s inequalities, itholds that | G | ≤ X p>Q b,h k b p k ∞ k∇ ˜ w p k X q ≤ p +2 λ sq k m q k . c r κ X p>Q b,h λ s +1 p k w p k X q ≤ p +2 λ s +1 q k m q k λ sq − p λ − q . c r κ X q ≥− ( λ s +2 q k w q k + λ s +2 q k m q k ) . Estimation of H. By Bony’s paraproduct decomposition, one has H = X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m p · ∇ v ≤ p − ) m q d x + X q ≥− X | p − q |≤ λ sq Z T ∆ q ( m ≤ p − · ∇ v p ) m q d x + X q ≥− X p ≥ q − λ sq Z T ∆ q ( ˜ m p · ∇ v p ) m q d x =: H + H + H . By the wavenumber Q u,v , the term H can be split into three parts. H = X − ≤ p ≤ Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( m p · ∇ v ≤ p − ) m q d x + X p>Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( m p · ∇ v ≤ Q u,v ) m q d x + X p>Q u,v +2 X | q − p |≤ λ sq Z T ∆ q ( m p · ∇ v ( Q u,v ,p − ) m q d x =: H + H + H . One can estimate H with the help of Definition 1.4, H¨older’s and Young’sinequalities. | H | ≤ X p ≥− k m p k rr − X | q − p |≤ λ sq k m q k X − ≤ p ′ ≤ p − λ p ′ k v p ′ k r . c r κ X p ≥− λ r p k m p k X | q − p |≤ λ sq k m q k X − ≤ p ′ ≤ p − λ − r p ′ . c r κ X p ≥− λ s +1 p k m p k X | q − p |≤ λ s +1 q k m q k X − ≤ p ′ ≤ p − λ − r p ′ − p . c r κ X q ≥− λ s +2 q k m q k . To estimate H , one recalls Definition 1.4 and applies H¨older’s and Young’sinequalities. | H | ≤k∇ v ≤ Q u,v k r X p>Q u,v k m p k rr − X | q − p |≤ λ sq k m q k . Λ − r u,v k v ≤ Q u,v k r X p>Q u,v λ − r p k m p k rr − X | q − p |≤ λ sq k m q k . c r κ X p ≥− λ s +1 p k m p k X | q − p |≤ λ s +1 q k m q k . c r κ X q ≥− λ s +2 q k m q k . As a result of Definition 1.4, H¨older’s, Young’s and Jensen’s inequalities, one has | H | ≤ X p>Q u,v +2 k m p k rr − X | q − p |≤ λ sq k m q k X Q u,v Q u,v +2 λ p k m p k X | q − p |≤ λ sq k m q k X Q u,v Q u,v X | q − p |≤ λ sq Z T ∆ q ( m ≤ p − · ∇ v p ) m q d x =: H + H , which are estimated by Definition 1.4, H¨older’s, Young’s and Jensen’s inequalities. | H | ≤ X − ≤ p ≤ Q u,v k∇ v p k r X | q − p |≤ λ sq k m q k X p ′ ≤ p − k m p ′ k rr − ≤ c r κ X − ≤ p ≤ Q u,v λ − r p X | q − p |≤ λ sq k m q k X p ′ ≤ p − λ r p ′ k m p ′ k ≤ c r κ X − ≤ p ≤ Q u,v X | q − p |≤ λ s +1 q k m q k X p ′ ≤ p − λ s +1 p ′ k m p ′ k λ r − s − p ′ − q . c r κ X q ≥− λ s +2 q k m q k ; ETERMINING WAVENUMBERS 33 | H | ≤ X p>Q u,v k∇ v p k r X | q − p |≤ λ sq k m q k X p ′ ≤ p − k m p ′ k rr − . X p>Q u,v λ − r p k v p k r X | q − p |≤ λ s +2 − r q k m q k X p ′ ≤ p − λ r p ′ k m p ′ k . c r κ X p>Q u,v X | q − p |≤ λ s +1 q k m q k X p ′ ≤ p − λ s +1 p ′ k m p ′ k λ s +1 − r q − p ′ . c r κ X q ≥− λ s +2 q k m q k . One also divides H into two terms. H = X − ≤ p ≤ Q u,v X q ≤ p +2 λ sq Z T ∆ q ( ˜ m p · ∇ v p ) m q d x + X p>Q u,v X q ≤ p +2 λ sq Z T ∆ q ( ˜ m p · ∇ v p ) m q d x =: H + H . By Definition 1.4, H¨older’s, Young’s and Jensen’s inequalities, one has | H | ≤ X − ≤ p ≤ Q u,v k∇ v p k r k ˜ m p k X q ≤ p +2 λ sq k m q k rr − . c r κ X − ≤ p ≤ Q u,v λ − r p k m p k X q ≤ p +2 λ s + r q k m q k . c r κ X − ≤ p ≤ Q u,v λ s +1 p k m p k X q ≤ p +2 λ s +1 q k m q k λ s − r q − p . c r κ X q ≥− λ s +2 q k m q k . For H , the following estimate holds. | H | ≤ X p>Q u,v k∇ v p k r k ˜ m p k X q ≤ p +2 λ sq k m q k rr − . c r κ X p>Q u,v λ − r p k m p k X q ≤ p +2 λ s + r q k m q k . c r κ X p>Q u,v λ s +1 p k m p k X q ≤ p +2 λ s +1 q k m q k λ s − r q − p . c r κ X q ≥− λ s +2 q k m q k . Conclusion. As the terms I and J are already estimated in Section 3, onesums up all the previous estimates and chooses a suitable constant c r to obtaindd t X q ≥− (cid:0) k w q k + k m q k (cid:1) . − X q ≥− λ q (cid:0) k w q k + k m q k (cid:1) . X q ≥− (cid:0) k w q k + k m q k (cid:1) . One can see that (cid:0) k w k L + k m k L (cid:1) decays to 0 exponentially as t → ∞ as a resultof Gr¨onwall’s inequality. (cid:3) Bounds on the wavenumbers In [11], it was shown that the time average of the determining wavenumber fora weak solution to the Navier-Stokes equations is bounded above by Kolmogorov’sdissipation wavenumber via the average energy dissipation rate ε := hk∇ u k L i , where h·i signifies the time average. For the 2D MHD system, it is also known thatexplicit dimension estimates of functional invariant sets can be given by the energydissipation rate.Yet, in the case of the Hall-MHD system, it seems impossible to bound thewavenumber Λ b,h ( t ) using the average magnetic energy dissipation rate hk∇ b k L i .Fortunately, restricting one’s attentions to strong solutions can lead to a reasonablebound on Λ b,h ( t ) in an average sense. Indeed, whenever Λ b,h ( t ) > λ , it must bethat one of the conditions in Definition 1.5 is unfulfilled, i.e., k b Q b,h ( t ) k ∞ > c r κ or k b ≤ Q b,h ( t ) − k ∞ > c r κ. The inequality k b Q b,h ( t ) k ∞ > c r κ implies thatΛ b,h ( t ) k b Q b,h ( t ) k ∞ > c r κ Λ b,h ( t ) . By Lemma 2.4, one has k∇ b k ∞ ≥ k∇ b Q b,h ( t ) k ∞ > (cid:0) c r κ Λ b,h ( t ) (cid:1) . Meanwhile, if k b ≤ Q b,h ( t ) − k ∞ > c r κ, thenΛ b,h ( t ) k b ≤ Q b,h ( t ) − k ∞ > c r κ Λ b,h ( t ) , which, by Lemma 2.4, yields k∇ b k ∞ ≥ k∇ b ≤ Q b,h ( t ) − k ∞ > (cid:0) c r κ Λ b,h ( t ) (cid:1) . Hence, for ( u, b ) ∈ L ∞ (cid:0) , ∞ ; ( H s ( T )) (cid:1) , one has, by Theorem 2.3, the followingbound. h Λ b,h i . k∇ b k L (0 ,T ; L ∞ ( R )) < ∞ . References [1] M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu. Kinetic formulation and global exis-tence for the Hall-Magneto-hydrodynamics system . Kinet. Relat. Models Vol. 4(4), 901-918,2011.[2] M. J. Benvenutti and L. C. F. Ferreira. Existence and stability of global large strong solutionsfor the Hall-MHD system. Differ. Integral Equ. Vol. 29(910), 9771000, 2016.[3] H. Bahouri, J. Chemin, and R. Danchin. 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Q b,h λ sp k∇ × m p k λ sq − p λ δQ b,h − p . c r µ X q ≥− λ sq k∇ m q k , | I | ≤ X q>Q b,h +2 λ sq k∇ × m q k X p ≥ q − k h p k ∞ k∇ × ˜ m p k . c r µ X q>Q b,h +2 λ sq k∇ × m q k X p ≥ q − λ sp k∇ × m p k λ sq − p λ δQ b,h − p . c r µ X q ≥− λ sq k∇ m q k . Thus, the estimation for I is completed. J , J and J remain to be estimated. One can write J , whose low frequencyparts vanish due to m ≤ Q b,h = 0, as J = X p>Q b,h +2 X | p − q |≤ λ sq Z T ∆ q (cid:0) m ( Q b,h ,p − × ( ∇ × b p ) (cid:1) · ( ∇ × m q )d x. Recalling Definition 1.5, one can estimate J using H¨older’s, Young’s and Jensen’sinequalities, provided that δ > s + 1 . | J | ≤ X p>Q b,h +2 λ p k b p k ∞ X | p − q |≤ λ sq k∇ × m q k X Q b,h
Q u,v k u p k r k∇ ˜ w p k X q ≤ p +2 λ sq k w q k rr − . c r ν X p>Q u,v λ − r p k w p k X q ≤ p +2 λ s + r q k w q k . c r ν X p>Q u,v λ s +1 p k w p k X q ≤ p +2 λ s +1 q k w q k λ s − r q − p . c r ν X q ≥− λ sq k∇ w q k . Estimation of B.
Q u,v λ r p k w p k X | q − p |≤ λ sq k w q k X p ′
Q u,v λ p k w p k X | q − p |≤ λ sq k w q k X p ′
Q u,v +2 k w p k X Q u,v
Q u,v +1 λ p k v p k r k w p k X q ≤ p +2 λ sq k w q k rr − . c r ν X p>Q u,v +1 λ − r p k w p k X q ≤ p +2 λ s + r q k w q k . c r ν X p>Q u,v +1 λ s +1 p k w p k X q ≤ p +2 λ s +1 q k w q k λ s − r q − p . c r ν X q ≥− λ sq k∇ w q k . Estimation of C.
Q b,h X Q b,h
− ( λ s +2 q k w q k + λ s +2 q k m q k ) . One uses H¨older’s, Young’s and Jensen’s inequalities to estimate F . | F | ≤ X p>Q b,h k∇ b p k ∞ k ˜ w p k X q ≤ p +2 λ sq k m q k ≤ c r κ X p>Q b,h λ p k w p k X q ≤ p +2 λ sq k m q k ≤ c r κ X p>Q b,h λ s +1 p k w p k X q ≤ p +2 λ s +1 q k m q k λ sq − p λ − q ≤ c r κ X q> − ( λ s +2 q k w q k + λ s +2 q k m q k ) . Estimation of G.