aa r X i v : . [ m a t h . A P ] F e b BLOW-UP OF DYADIC MHD MODELS WITH FORWARDENERGY CASCADE
MIMI DAI
Abstract.
A particular type of dyadic model for the magnetohydrodynamics(MHD) with forward energy cascade is studied. The model includes intermit-tency dimension δ in the nonlinear scales. It is shown that when δ is small,positive solution with large initial data for either the dyadic MHD model orthe dyadic Hall MHD develops blow-up in finite time.KEY WORDS: magnetohydrodynamics; Hall effect; intermittency; dyadicmodel; energy cascade; blow-up.CLASSIFICATION CODE: 35Q35, 76D03, 76W05. Introduction
Dyadic models for the incompressible magnetohydrodynamics (MHD) with Halleffect governed by u t + u · ∇ u − B · ∇ B + ∇ p = ν ∆ u,B t + u · ∇ B − B · ∇ u + d i ∇ × (( ∇ × B ) × B ) = µ ∆ B, ∇ · u = 0 , (1.1)were derived in [12], where intermittency effect enters the derivation in a naturalway. In system (1.1), the unknown functions u , p and B denote respectively theelectrically conducting fluid velocity field, fluid pressure, and magnetic field influ-enced by the conducting fluid. The parameters ν, µ and d i stand for the kinematicviscosity, magnetic resistivity and ion inertial length, respectively. We assume (1.1)is posed either on R × [0 , ∞ ) or T × [0 , ∞ ) . A general form of the derived dyadic(shell) model for (1.1) reads as ddt a j + νλ j a j + α (cid:18) λ − δu j a j a j +1 − λ − δu j − a j − (cid:19) + β (cid:18) λ − δu j a j +1 − λ − δu j − a j − a j (cid:19) + α (cid:18) λ − δb j b j b j +1 − λ − δb j − b j − (cid:19) + β (cid:18) λ − δb j +1 b j +1 − λ − δb j b j − b j (cid:19) = 0 , (1.2) The author was partially supported by NSF grants DMS–1815069 and DMS–2009422. ddt b j + µλ j b j + α (cid:18) λ − δb j a j b j +1 − λ − δb j − a j − b j − (cid:19) + β (cid:18) λ − δb j +1 a j +1 b j +1 − λ − δb j a j b j − (cid:19) + α (cid:18) λ − δb j b j a j +1 − λ − δb j − a j − b j − (cid:19) + β (cid:18) λ − δb j +1 b j +1 a j +1 − λ − δb j a j − b j (cid:19) + d i α (cid:18) λ − δb j b j b j +1 − λ − δb j − b j − (cid:19) + d i β (cid:18) λ − δb j b j +1 − λ − δb j − b j b j − (cid:19) = 0 , (1.3)for j ≥ , which is an ODE system of infinitely many equations. In system (1.2)-(1.3), the unknown functions a j and b j appear to be the kinetic energy and magneticenergy in the j -th shell, respectively, in the derivation. However, they can also betreated as Fourier coefficients of u and B , respectively. By convention, we take a = b = 0 . The parameter λ j = λ j stands for the wavenumber of the j -th shellfor some λ > . The parameters δ u and δ b represent intermittency dimension forthe velocity field u and magnetic field B , respectively, which are defined throughthe saturation level of Bernstein’s inequality, see [7, 12]. To be physically relevant, δ u and δ b take values in [0 , . The situation of δ u = δ b = 3 corresponds to theKolmogorov regime, in which case both of the conducting flow and magnetic fieldflow are homogeneous, isotropic and self-similar. In the case of δ u = δ b = 0 , bothflows are extremely inhomogeneous and singular. The parameters α k and β k for ≤ k ≤ play essential roles in interpreting energy transfer among shells and thecoupling relationship between the velocity field and magnetic field. They will befurther discussed at a later time.The dyadic model (1.2)-(1.3) is derived under the following principles: (i) kineticenergy and magnetic energy are balanced through each shell; (ii) the total energyis conserved when ν = µ = 0 ; (iii) only local interactions among shells are takeninto account (in fact, only interactions with the first neighbor shells are employedhere). One can check that the total energy E ( t ) = 12 X j ≥ (cid:0) a j ( t ) + b j ( t ) (cid:1) (1.4)is indeed formally conserved for the model with ν = µ = 0 and any parameters α k and β k , ≤ k ≤ . Moreover, the total energy is also formally conserved forthe system with: (i) α k = 0 and β k = 0 for ≤ k ≤ , in which case the dyadicmodel is the Obukov type; (ii) β k = 0 and α k = 0 for ≤ k ≤ , in which casethe dyadic model is the Katz-Pavlóvic (KP) type, see [19, 20]. It is importantto notice that the sign of the parameters α k and β k determines the direction ofenergy transfer: positive sign indicates forward energy cascade, while negative signindicates backward energy cascade.Dyadic models for hydrodynamics governed by the Navier-Stokes equation (NSE)and Euler equation have been extensively studied, for instance, see [1, 2, 3, 4, 6, 8,9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25]. (It would be quite challengingto list the complete literature on this topic; thus, the author gives up such attempthere.) They serve as approximating models for the true fluid flows, which reflectsome most essential features of the turbulent flows. In fact, taking b j = 0 for j ≥ , α = 1 and β = 0 in (1.2), the model reduces to the KP dyadic model; instead,taking b j = 0 for j ≥ , α = 0 and β = 1 in (1.2) makes it to be the Obukovmodel. One major shortage of these dyadic models is that spatial complexity and LOW-UP OF DYADIC MHD MODELS 3 geometry structures of the original flows are over simplified. Nevertheless, thestudy of dyadic models has provided important insights in the understanding ofhydrodynamic turbulence.Dyadic models for the MHD turbulence were also introduced and studied byphysicists, see [5, 17], the recent article [24] and references therein. The dyadicmodel (1.2)-(1.3), derived based on harmonic analysis techniques and with inter-mittency effect included automatically, recovers some models from the physics com-munity which will be pointed out below at the proper place. The main aim ofproposing model (1.2)-(1.3) is two-fold: (i) understand how the behaviours of solu-tions depend on the intermittency effect; (ii) explore how different energy cascadeand coupling relationships affect the dynamics.In [12], the questions of well-posedness and finite-time blow-up were addressedfor a special case of the model (1.2)-(1.3). The particular model is obtained bytaking the parameters α = α = α = 1 , α = − , β k = 0 with ≤ k ≤ , and δ u = δ b := δ in (1.2)-(1.3). Namely, the following model was considered, ddt a j + νλ j a j = − (cid:0) λ θj a j a j +1 − λ θj − a j − (cid:1) + (cid:0) λ θj b j b j +1 − λ θj − b j − (cid:1) ,ddt b j + µλ j b j = − (cid:0) λ θj a j b j +1 − λ θj b j a j +1 (cid:1) − d i (cid:0) λ θ +1 j b j b j +1 − λ θ +1 j − b j − (cid:1) , (1.5)for j ≥ , a = b = 0 , and θ = − δ . Some important features about this modelare described below. First, the total energy E ( t ) as in (1.4) is formally conservedin the inviscid non-resistive case, i.e. ν = µ = 0 ; the cross helicity defined by H c ( t ) = X j ≥ a j ( t ) b j ( t ) (1.6)is also formally conserved in the inviscid non-resistive MHD case, i.e. ν = µ = d i =0 . Second, from the signs of the nonlinear terms, we observe that energy movestowards larger frequency (forward energy cascade) within the nonlinear structuresof ( u · ∇ ) u and the Hall effect ∇ × (( ∇ × B ) × B ) ; energy moves toward smallerfrequency (backward energy cascade) within the coupled nonlinear structures. Thefollowing diagram illustrate the energy transfer among neighbor shells for system(1.5), · · · −→ a j − −→ a j −→ a j +1 −→ · · ·↑ ւ ↑ ւ ↑· · · −→ b j − −→ b j −→ b j +1 −→ · · · Third, when δ = 3 and hence θ = 1 , this model corresponds to the so called L1model derived by physicists, see [17, 24]. The name L1 means that each flux termhas local, two feet in the same shell and the third foot in a neighboring shell, forinstance, λ θj a j a j +1 and λ θj a j b j b j +1 .For system (1.5), existence of global in time weak solutions is obtained in [12]for any δ ∈ [0 , (and any θ > ); when d i > , strong solution is shown to existlocally for δ ∈ (1 , and globally for δ = 3 ; while for d i = 0 , strong solution canbe obtained locally for δ ∈ [0 , and globally for δ ∈ [1 , . Moreover, when d i > and δ < − , positive solution of (1.5) with large initial data is shown to developblow-up at finite time. However, the last scenario is physically irrelevant since theintermittency dimension δ is only physically meaningful if δ ∈ [0 , . We point outthat the question of whether positive solution of (1.5) with d i = 0 (i.e. the MHDdyadic model) develops blow-up at finite time remains open. LOW-UP OF DYADIC MHD MODELS 4
In the current paper, we will work with another particular case of the generaldyadic model (1.2)-(1.3) with only forward energy cascade. Specifically, we assume δ u = δ b = δ and denote θ = − δ as before, take α = α = α = 1 , α = − , β k = 0 with ≤ k ≤ , and consider the following model ddt a j + νλ j a j = − (cid:0) λ θj a j a j +1 − λ θj − a j − (cid:1) − (cid:0) λ θj b j b j +1 − λ θj − b j − (cid:1) ,ddt b j + µλ j b j = (cid:0) λ θj a j b j +1 − λ θj b j a j +1 (cid:1) − d i (cid:0) λ θ +1 j b j b j +1 − λ θ +1 j − b j − (cid:1) , (1.7)for j ≥ and a = b = 0 . An obvious difference between system (1.5) and system(1.7) is the sign of the coupling terms (cid:0) λ θj b j b j +1 − λ θj − b j − (cid:1) and (cid:0) λ θj a j b j +1 − λ θj b j a j +1 (cid:1) .That leads to some more sophisticated different features. For system (1.7), althoughthe total energy is still formally conserved if ν = µ = 0 , the cross helicity as definedin (1.6) is no longer conserved with ν = µ = d i = 0 . Another important feature isthat there is only forward energy cascade within the dynamics, see the illustrationbelow · · · −→ a j − −→ a j −→ a j +1 −→ · · ·↓ ր ↓ ր ↓· · · −→ b j − −→ b j −→ b j +1 −→ · · · The existence of short time strong solution and global strong solution to (1.7)can be established for proper regimes of the intermittency dimension δ , in a similarfashion as the analysis for (1.5) in [12]. In this paper, we pursue to construct finite-time blow-up solutions to (1.7) with either d i = 0 or d i > when the intermittencydimension is below certain threshold.In the case of the MHD dyadic model with θ > , we will show that blow-updevelops at finite time for positive solution with large initial data in the space H s × H s with s > θ . Theorem 1.1.
Let ( a ( t ) , b ( t )) be a positive solution to (1.7) with d i = 0 and θ > . Let λ ≥ . For any γ > , there exists a constant M such that if k a (0) k γ + k b (0) k γ > M , then k a ( t ) k θ + γ + k b ( t ) k θ + γ is not locally integrable on [0 , ∞ ) . On the other hand, for the Hall MHD dyadic model with θ > , finite timeblow-up occurs for positive solution with large initial data in the space H s × H + s with s > θ . Theorem 1.2.
Let ( a ( t ) , b ( t )) be a positive solution to (1.7) with d i > and θ > .For any γ > , there exists a constant M such that if k a (0) k γ + k b (0) k γ > M ,then k a ( t ) k θ + γ + k b ( t ) k ( θ +1)+ γ is not locally integrable on [0 , ∞ ) . Remark 1.3.
Since θ = − δ , θ > is equivalent to δ < − . Remark 1.4.
In Theorem 1.1, the parameter λ , the basis of the wavenumber λ j = λ j , can be taken as any value larger than 1. To reduce the complexity ofanalyzing parameters satisfying (4.29)-(4.34), we choose λ ≥ . Remark 1.5.
The question whether a solution of (1.7) with positive initial dataremains positive is open and will be addressed in future investigation. It is knownthat, if B = 0 and hence b j = 0 for all j ≥ , the reduced NSE dyadic model (1.7)with positive initial data produces positive solutions, see [6]. LOW-UP OF DYADIC MHD MODELS 5
Remark 1.6.
In view of the fact that the Hall MHD system (1.1) with d i > involves a more singular nonlinear structure of the Hall effect, reflected in thedyadic model (1.2)-(1.3) with a larger nonlinear scale d i (cid:0) λ θ +1 j b j b j +1 − λ θ +1 j − b j − (cid:1) ,one might expect to show blow-up for system (1.7) with d i > for smaller θ , thatis, for θ < . Nevertheless, in the proof of Theorem 1.2 in Section 5, it appears thatthe coupling terms cause serious barrier to lower the threshold of θ for blow-up.That could be just the limitation of the approach of proving blow-up in this paper.There is hope to move down the threshold of θ for blow-up by other frameworks ofproving blow-up.An interesting connection between the intermittency effect and dissipation strengthcan be revealed through dyadic models in the following way. In fact, the dyadicsystem (1.7) can be rescaled to ddt a j + ν ¯ λ αj a j = − ¯ λ j a j a j +1 + ¯ λ j − a j − − ¯ λ j b j b j +1 + ¯ λ j − b j − ,ddt b j + µ ¯ λ αj b j = ¯ λ j a j b j +1 − ¯ λ j b j a j +1 − d i (cid:0) ¯ λ α +1 j b j b j +1 − ¯ λ α +1 j − b j − (cid:1) (1.8)with α = 1 θ = 25 − δ , by rescaling the wavenumber λ j = ¯ λ αj . System (1.8) can be seen as the dyadicmodel of the Hall-MHD system with generalized diffusions ( − ∆) α u and ( − ∆) α B .The results of Theorem 1.1 and Theorem 1.2 can be transformed to system (1.8)as follows. Corollary 1.7.
Let ( a ( t ) , b ( t )) be a positive solution to (1.8) with d i = 0 and α < .For any γ > , there exists a constant M such that if k a (0) k γ + k b (0) k γ > M ,then k a ( t ) k + γ + k b ( t ) k + γ is not locally integrable on [0 , ∞ ) . Corollary 1.8.
Let ( a ( t ) , b ( t )) be a positive solution to (1.8) with d i > and α < .For any γ > , there exists a constant M such that if k a (0) k γ + k b (0) k γ > M ,then k a ( t ) k + γ + k b ( t ) k ( α +1)+ γ is not locally integrable on [0 , ∞ ) .The proof of Theorem 1.1 and Theorem 1.2 relies on a contradiction argumentand the construction of a Lyapunov function L ( t ) which would satisfy a Riccatitype of inequality. Depending on whether d i > or not, i.e. whether the Hall termis present or not, the choice of L ( t ) is different. The construction of L ( t ) for boththe dyadic MHD and Hall MHD models is described in Section 3; some propertiesof L ( t ) are also established there. The proof of Theorem 1.1 and Theorem 1.2 isprovided in Section 4 and Section 5, respectively. On the other hand, Corollary 1.7and Corollary 1.8 can be justified automatically from the rescaling relationship.2. Notations and notion of solutions
We denote H = l which is endowed with the standard scalar product and norm, ( u, v ) := ∞ X n =1 u n v n , | u | := p ( u, u ) . As mentioned earlier, we choose the wavenumber λ n = λ n for a constant λ > , andall integers n ≥ . Corresponding to the standard Sobolev space H s for functions LOW-UP OF DYADIC MHD MODELS 6 with spacial variables, we use the same notation H s here to represent the space fora sequence { u n } ∞ n =1 , which is endowed with the scaler product ( u, v ) s := ∞ X n =1 λ sn u n v n and the norm k u k s := p ( u, u ) s . We notice that H = H = l which is regarded as the energy space.In the following, the concept of solutions for the dyadic system (1.7) is intro-duced. Definition 2.1.
A pair of H -valued functions ( a ( t ) , b ( t )) defined on [ t , ∞ ) is saidto be a weak solution of (1.7) if a j and b j satisfy (1.7) and a j , b j ∈ C ([ t , ∞ )) forall j ≥ . Definition 2.2.
A solution ( a ( t ) , b ( t )) of (1.7) is strong on [ T , T ] if k a k and k b k are bounded on [ T , T ] . A solution is strong on [ T , ∞ ) if it is strong on everyinterval [ T , T ] for any T > T .3. Lyapunov function and auxiliary estimates
In this section, we construct a Lyapunov function for system (1.7) and presentits continuity under certain conditions. In particular, if d i = 0 , we consider L ( t ) := k a ( t ) k γ + k b ( t ) k γ + c ∞ X j =1 λ γj a j ( t ) a j +1 ( t )+ c ∞ X j =1 λ γj b j ( t ) a j +1 ( t ) + c ∞ X j =1 λ γj a j ( t ) b j ( t ) (3.9)for some appropriate positive constants c , c , and c . The main principle of de-signing L ( t ) is to have terms a j and b j included in ddt L ( t ) , which will play a crucialrole to derive a Riccati type of inequality for L ( t ) . In fact, ddt ( a j a j +1 ) produces λ θj a j and ddt ( b j a j +1 ) gives λ θj b j . However, it turns out that the term λ θj b j is notenough to control a flux triple term λ θj b j b j +1 b j +2 in the estimates. It is the reasonthat we include the term λ γj a j b j in L ( t ) , and hence ddt ( a j b j ) gives a term λ θj b j b j +1 which can contribute to control λ θj b j b j +1 b j +2 .For the dyadic Hall MHD model (1.7) with d i > , we choose L ( t ) := k a ( t ) k γ + k b ( t ) k γ + c ∞ X j =1 λ γj a j ( t ) a j +1 ( t )+ c ∞ X j =1 λ γj b j ( t ) b j +1 ( t ) (3.10)for appropriate constants c > and c > . As for the MHD case, ddt ( a j a j +1 ) includes the good term λ θj a j . While in this case, ddt ( b j b j +1 ) contributes a goodterm λ θ +1 j b j due to the presence of the Hall term; in the same time, ddt b j (from ddt k b ( t ) k γ ) gives λ θ +1 j b j b j +1 , also due to the Hall effect. Thus, the two good termstogether are able to control many negative flux terms including λ θ +1 j b j b j +1 b j +2 . LOW-UP OF DYADIC MHD MODELS 7
In the rest of this section, we will provide some auxiliary estimates and show thecontinuity of L ( t ) under certain conditions on the solution. Lemma 3.1. (i) If θ > γ , there exists a constant c > such that ∞ X j =1 λ γ + θj a j ≥ c k a k γ +1 , ∞ X j =1 λ γ + θj b j ≥ c k b k γ +1 , ∞ X j =1 λ γ + θ +1 j b j ≥ c k b k γ +1 . (ii) If θ > γ , we also have k a ( t ) k γ +1 ≤ k a ( t ) k θ + γ , k b ( t ) k γ +1 ≤ k b ( t ) k θ + γ . (iii) The following inequalities ∞ X j =1 λ γ +2 j a j a j +1 ≤ λ − γ − k a k γ +1 ∞ X j =1 λ γ +2 j b j b j +1 ≤ λ − γ − k b k γ +1 ∞ X j =1 λ γ +2 j a j b j ≤ (cid:0) k a k γ +1 + k b k γ +1 (cid:1) ∞ X j =1 λ γ +2 j b j a j +1 ≤ λ − γ − (cid:0) k a k γ +1 + k b k γ +1 (cid:1) hold.(iv) For positive a j and b j with j ≥ , we have ∞ X j =1 λ γ + θj a j a j +1 ≤ k a k θ + γ , ∞ X j =1 λ γ + θj b j b j +1 ≤ k b k θ + γ , ∞ X j =1 λ γ + θj b j a j +1 ≤ k a k θ + γ + k b k θ + γ . Proof:
The justification of the inequalities in (i) is rather standard and thusomitted here. One can find a quick proof in [12]. The inequalities in (ii) followimmediately from the fact θ > γ and hence γ + 1 < θ + γ . The ones in (iii)are not complicated either and we only show one of them below. Applying Hölder’s LOW-UP OF DYADIC MHD MODELS 8 and Young’s inequalities, we have ∞ X j =1 λ γ +2 j b j a j +1 = λ − γ − ∞ X j =1 (cid:16) λ γ +1 j b j (cid:17) (cid:16) λ γ +1 j +1 a j +1 (cid:17) ≤ λ − γ − ∞ X j =1 λ γ +2 j b j ∞ X j =1 λ γ +2 j +1 a j +1 ≤ λ − γ − (cid:0) k a k γ +1 + k b k γ +1 (cid:1) . We only show the last inequality of (iv); another two can be proved similarly.The application of Young’s inequality and a basic inequality for sum leads to ∞ X j =1 λ γ + θj b j a j +1 ≤ ∞ X j =1 λ γ + θj (cid:18) b j + 13 a j +1 (cid:19) ≤ ∞ X j =1 λ γ + θj a j + ∞ X j =1 λ γ + θj b j ≤ ∞ X j =1 λ (2 γ + θ ) j a j + ∞ X j =1 λ (2 γ + θ ) j b j ≤k a k θ + γ + k b k θ + γ . (cid:3) Lemma 3.2.
Let ( a ( t ) , b ( t )) be a positive solution to (1.7) with d i = 0 . Assume k a ( t ) k θ + γ + k b ( t ) k θ + γ is locally integrable on [0 , ∞ ) . Then L ( t ) defined in(3.9) is continuous on [0 , ∞ ) . Proof:
We denote E γ ( t ) := k a ( t ) k γ + k b ( t ) k γ ,f ( t ) := c ∞ X j =1 λ γj a j ( t ) a j +1 ( t ) + c ∞ X j =1 λ γj b j ( t ) a j +1 ( t ) + c ∞ X j =1 λ γj a j ( t ) b j ( t ) . Under the assumption, we show that both E γ and f are continuous on [0 , ∞ ) .Applying the two equations of (1.7) with d i = 0 , and taking the sum for all j ≥ , we find that E γ ( t ) − E γ (0)= − Z t ν k a ( τ ) k γ +1 + µ k b ( τ ) k γ +1 dτ + 2( λ γ − Z t ∞ X j =1 λ γ + θj a j a j +1 dτ + 2( λ γ − Z t ∞ X j =1 λ γ + θj b j a j +1 dτ. Combining the inequalities of Lemma 3.1 (ii) and (iv) and the assumption that k a ( t ) k θ + γ + k b ( t ) k θ + γ is locally integrable, we conclude that k a ( t ) k γ +1 , k b ( t ) k γ +1 , P ∞ j =1 λ γ + θj a j a j +1 , and P ∞ j =1 λ γ + θj b j a j +1 are all locally integrable as well. There-fore, the integrals on the right hand side of the equation above are all defined forany t > . It thus follows that E γ is continuous on [0 , ∞ ) . LOW-UP OF DYADIC MHD MODELS 9
We denote for j ≥ f j ( t ) = c λ γj a j ( t ) a j +1 ( t ) + c λ γj b j ( t ) a j +1 ( t ) + c λ γj a j ( t ) b j ( t ) . For any t > , we infer lim sup t → t | f ( t ) − f ( t ) | = lim sup t → t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =1 f j ( t ) − ∞ X j =1 f j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim J →∞ lim sup t → t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J − X j =1 f j ( t ) − J − X j =1 f j ( t ) + ∞ X j = J f j ( t ) − ∞ X j = J f j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim J →∞ lim sup t → t J − X j =1 | f j ( t ) − f j ( t ) | + lim J →∞ lim sup t → t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = J f j ( t ) − ∞ X j = J f j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.11)According to the definition of solution in Definition 2.1, f j is continuous for any j ≥ and hence lim t − t | f j ( t ) − f j ( t ) | = 0 , ∀ ≤ j ≤ J − . It implies that lim J →∞ lim sup t → t J − X j =1 | f j ( t ) − f j ( t ) | = 0 . (3.12)To analyze the last limit in (3.11), we observe that from Lemma 3.1 (iii) ≤ f ( t ) ≤ c k a ( t ) k γ + 4 c k b ( t ) k γ ≤ c + c ) E γ ( t ) . The continuity of E γ on [0 , ∞ ) implies f is bounded on every interval [ T , T ] , forany T > T ≥ . Therefore, it follows that lim J →∞ lim sup t → t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = J f j ( t ) − ∞ X j = J f j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (3.13)In view of (3.11)-(3.13), we claim f is continuous on [0 , ∞ ) . It accomplishes theproof of the lemma. (cid:3) When d i > , we have the following statement. Lemma 3.3.
Let ( a ( t ) , b ( t )) be a positive solution to (1.7) with d i > . Assume k a ( t ) k θ + γ + k b ( t ) k ( θ +1)+ γ is locally integrable on [0 , ∞ ) . Then L ( t ) defined in(3.10) is continuous on [0 , ∞ ) . Proof:
The proof follows a close line to that of Lemma 3.2. We only explain whyit requires k b ( t ) k ( θ +1)+ γ to be locally integrable on [0 , ∞ ) . Indeed, multiplyingthe a j equation of (1.7) by λ γj a j and the b j equation with d i > by λ γj b j , adding LOW-UP OF DYADIC MHD MODELS 10 all the shells for j ≥ , and integrating from to t , we obtain E γ ( t ) − E γ (0)= − Z t ν k a ( τ ) k γ +1 + µ k b ( τ ) k γ +1 dτ + 2( λ γ − Z t ∞ X j =1 λ γ + θj a j a j +1 dτ + 2 d i ( λ γ − Z t ∞ X j =1 λ γ + θ +1 j b j b j +1 dτ + 2( λ γ − Z t ∞ X j =1 λ γ + θj b j a j +1 dτ. Referring to the second inequality of Lemma 3.1 (iv) with θ replaced by θ + 1 ,the assumption k b ( t ) k ( θ +1)+ γ is locally integrable on [0 , ∞ ) guarantees that P ∞ j =1 λ γ + θ +1 j b j b j +1 is locally integrable on [0 , ∞ ) . (cid:3) Blow-up of positive solutions of dyadic MHD
This section is devoted to a proof of Theorem 1.1. The following lemma playsan important role, whose proof is postponed to a later time.
Lemma 4.1.
Consider system (1.7) with d i = 0 . Let θ > and < γ < θ − . Fix λ ≥ . Assume k a (0) k γ + k b (0) k γ > M for a certain constant M > . Then, thefunction L ( t ) defined in (3.9) for positive solution ( a ( t ) , b ( t )) of (1.7) is a Lyapunovfunction and it blows up in finite time. Proof of Theorem 1.1:
We adapt a contradiction argument here. Supposethat ( a ( t ) , b ( t )) is a positive solution to (1.7) with d i = 0 such that k a ( t ) k θ + γ + k b ( t ) k θ + γ is integrable on [0 , T ] for any T > , with γ ∈ (0 , θ − and γ ≪ .It follows from Lemma 3.2 that L defined in (3.9) is continuous on [0 , ∞ ) . Theassumption of k a (0) k γ + k b (0) k γ > M for a constant M > along with Lemma4.1 implies that the function L blows up in finite time. Obviously, the last twoproperties of L leads to a contradiction.We also notice that k a ( t ) k θ + γ + k b ( t ) k θ + γ is not locally integrable for anarbitrarily small γ > implies it is not locally integrable for any γ > . (cid:3) We are left to give a justification of Lemma 4.1.
Proof of Lemma 4.1:
The goal is to show that for some
T > , L ( t ) > L (0) , ∀ t ∈ (0 , T ] , and L satisfies a Riccati type of inequality.Utilizing the two equations of (1.7) with d i = 0 , straightforward computationshows that ddt (cid:16) λ γj a j a j +1 (cid:17) = − ν (1 + λ ) λ γ +2 j a j a j +1 + λ γ + θj a j + λ γ + θj a j b j + λ θj − λ γj a j − a j +1 + λ θj − λ γj b j − a j +1 − λ γ + θj a j a j +1 − λ γj λ θj +1 a j a j +1 a j +2 − λ γ + θj b j a j +1 b j +1 − λ γj λ θj +1 a j b j +1 b j +2 , (4.14) LOW-UP OF DYADIC MHD MODELS 11 ddt (cid:16) λ γj b j a j +1 (cid:17) = − ( µ + νλ ) λ γ +2 j b j a j +1 + λ γ + θj b j + λ γ + θj a j b j + λ γ + θj a j a j +1 b j +1 − λ γ + θj b j a j +1 − λ γj λ θj +1 b j a j +1 a j +2 − λ γj λ θj +1 b j b j +1 b j +2 , (4.15) ddt X j ≥ λ γj a j b j = − ( ν + µ ) X j ≥ λ γ +2 j a j b j + (cid:0) λ γ + 1 (cid:1) X j ≥ λ γ + θj a j b j +1 + (cid:0) λ γ − (cid:1) X j ≥ λ γ + θj b j b j +1 − X j ≥ λ γ + θj a j b j a j +1 . (4.16)In the same time, we have the energy equality ddt (cid:0) k a ( t ) k γ + k b ( t ) k γ (cid:1) = − ν k a ( t ) k γ +1 − µ k b ( t ) k γ +1 + 2( λ γ − ∞ X j =1 λ γ + θj a j a j +1 + 2( λ γ − ∞ X j =1 λ γ + θj b j a j +1 . (4.17)The task is to control the negative terms on the right hand side of (4.14)-(4.17)using the positive terms λ γ + θj a j , λ γ + θj a j b j , λ γ + θj b j , λ γ + θj a j a j +1 , λ γ + θj b j a j +1 and λ γ + θj b j b j +1 . We estimate these negative terms by applying Young’s inequalityas follows, λ γ + θj a j a j +1 = λ − (2 γ + θ ) (cid:16) λ (2 γ + θ ) j a j a j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +1 a j +1 (cid:17) ≤ λ − (2 γ + θ ) λ γ + θj +1 a j +1 + 12 λ − (2 γ + θ ) λ γ + θj a j a j +1 ; (4.18) λ γj λ θj +1 a j a j +1 a j +2 = λ γj λ θj +1 (cid:16) a j a j +1 (cid:17) (cid:16) a j +1 a j +2 (cid:17) (cid:16) a j +2 (cid:17) ≤ λ θ λ γ + θj a j a j +1 + 14 λ − γ λ γ + θj +1 a j +1 a j +2 + 14 λ − γ − θ λ γ + θj +2 a j +2 ; (4.19) λ γ + θj b j a j +1 b j +1 = λ − (2 γ + θ ) (cid:16) λ (2 γ + θ ) j b j a j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +1 a j +1 b j +1 (cid:17) ≤ λ − (2 γ + θ ) λ γ + θj b j a j +1 + 12 λ − (2 γ + θ ) λ γ + θj +1 a j +1 b j +1 ; (4.20) λ γj λ θj +1 a j b j +1 b j +2 = λ − γ (cid:16) λ (2 γ + θ ) j a j (cid:17) (cid:16) λ (2 γ + θ ) j +1 b j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +2 b j +2 (cid:17) ≤ λ − γ λ γ + θj a j + 13 λ − γ λ γ + θj +1 b j +1 + 13 λ − γ λ γ + θj +2 b j +2 ; (4.21) LOW-UP OF DYADIC MHD MODELS 12 λ γ + θj b j a j +1 = λ − (2 γ + θ ) (cid:16) λ (2 γ + θ ) j b j a j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +1 a j +1 (cid:17) ≤ λ − (2 γ + θ ) λ γ + θj b j a j +1 + 12 λ − (2 γ + θ ) λ γ + θj +1 a j +1 ; (4.22) λ γj λ θj +1 b j a j +1 a j +2 = λ γj λ θj +1 (cid:16) b j a j +1 (cid:17) (cid:16) a j +1 a j +2 (cid:17) (cid:16) a j +2 (cid:17) ≤ λ θ λ γ + θj b j a j +1 + 14 λ − γ λ γ + θj +1 a j +1 a j +2 + 14 λ − γ − θ λ γ + θj +2 a j +2 ; (4.23) λ γj λ θj +1 b j b j +1 b j +2 = λ γj λ θj +1 (cid:16) b j b j +1 (cid:17) (cid:16) b j +1 b j +2 (cid:17) (cid:16) b j +2 (cid:17) ≤ λ θ λ γ + θj b j b j +1 + 14 λ − γ λ γ + θj +1 b j +1 b j +2 + 14 λ − γ − θ λ γ + θj +2 b j +2 ; (4.24) λ γ + θj a j b j a j +1 = 2 λ γ + θj ( a j a j +1 )( b j a j +1 ) ≤ λ γ + θj a j a j +1 + λ γ + θj b j a j +1 . (4.25)Applying (4.18), (4.19), (4.20) and (4.21) to (4.14), multiplying the constant c i ,and adding the shells for j ≥ , we obtain ddt c ∞ X j =1 λ γj a j a j +1 ≥ − νc (1 + λ ) ∞ X j =1 λ γ +2 j a j a j +1 + c (cid:18) − λ − (2 γ + θ ) − λ − γ − θ − λ − γ (cid:19) ∞ X j =1 λ γ + θj a j − c λ − γ ∞ X j =1 λ γ + θj b j + c (cid:18) − λ − (2 γ + θ ) (cid:19) ∞ X j =1 λ γ + θj a j b j − c (cid:18) λ − (2 γ + θ ) + 12 λ θ + 14 λ − γ (cid:19) ∞ X j =1 λ γ + θj a j a j +1 − c λ − (2 γ + θ ) ∞ X j =1 λ γ + θj b j a j +1 . (4.26) LOW-UP OF DYADIC MHD MODELS 13
Similarly putting (4.15) together with (4.22), (4.23) and (4.24) gives rise to ddt c ∞ X j =1 λ γj b j a j +1 ≥ − ( µ + νλ ) c ∞ X j =1 λ γ +2 j b j a j +1 + c ∞ X j =1 λ γ + θj a j b j + c (cid:18) − λ − γ − θ (cid:19) ∞ X j =1 λ γ + θj b j − c (cid:18) λ − (2 γ + θ ) + 14 λ − γ − θ (cid:19) ∞ X j =1 λ γ + θj a j − c (cid:16) λ − (2 γ + θ ) + λ θ (cid:17) ∞ X j =1 λ γ + θj b j a j +1 − c λ − γ ∞ X j =1 λ γ + θj a j a j +1 − c (cid:18) λ θ + 14 λ − γ (cid:19) ∞ X j =1 λ γ + θj b j b j +1 . (4.27)In the end, (4.16) along with (4.25) implies ddt c X j ≥ λ γj a j b j ≥ − ( ν + µ ) c X j ≥ λ γ +2 j a j b j + c (cid:0) λ γ + 1 (cid:1) X j ≥ λ γ + θj a j b j +1 + c (cid:0) λ γ − (cid:1) X j ≥ λ γ + θj b j b j +1 − c ∞ X j =1 λ γ + θj a j a j +1 − c ∞ X j =1 λ γ + θj b j a j +1 . (4.28)Comparing the coefficients of P ∞ j =1 λ γ + θj a j , P ∞ j =1 λ γ + θj b j , P ∞ j =1 λ γ + θj a j b j , P ∞ j =1 λ γ + θj a j a j +1 , P ∞ j =1 λ γ + θj b j a j +1 and P ∞ j =1 λ γ + θj b j b j +1 on the right handside of (4.17) and (4.26)-(4.28), we impose the following conditions for a constant c > c (cid:18) − λ − (2 γ + θ ) − λ − γ − θ − λ − γ (cid:19) − c (cid:18) λ − (2 γ + θ ) + 14 λ − γ − θ (cid:19) ≥ c , (4.29) c (cid:18) − λ − γ − θ (cid:19) − c λ − γ ≥ c , (4.30) c (cid:18) − λ − (2 γ + θ ) (cid:19) ≥ , (4.31) λ γ − − c (cid:18) λ − (2 γ + θ ) + 12 λ θ + 14 λ − γ (cid:19) − c λ − γ − c ≥ , (4.32) LOW-UP OF DYADIC MHD MODELS 14 λ γ − − c λ − (2 γ + θ ) − c (cid:16) λ − (2 γ + θ ) + λ θ (cid:17) − c ≥ , (4.33) c ( λ γ − − c (cid:18) λ θ + 14 λ − γ (cid:19) ≥ . (4.34)We can choose < c = c ≪ c ≪ , such that there exists a constant c > withthe conditions (4.29)-(4.34) satisfied for θ > , λ ≥ and any γ ∈ (0 , − θ ) . Indeed,we observe that: condition (4.31) is automatically satisfied; (4.32) and (4.33) aresatisfied provided c ≪ c , c ≤ λ γ − λ θ + λ − γ ; while (4.34) is satisfied if c ≤ c ( λ γ − λ θ + λ − γ ; in the end, we can choose c = c and c = min (cid:26) c (cid:18) − λ − (2 γ + θ ) − λ − γ − θ − λ − γ (cid:19) , c (cid:18) − λ − γ − θ − λ − γ (cid:19)(cid:27) which makes (4.29) and (4.30) valid.For the constants c , c , c and c chosen above, we add (4.17) and (4.26)-(4.28)to infer ddt L ( t ) ≥ − ν (1 + λ ) ∞ X j =1 λ γ +2 j a j a j +1 − µ (1 + λ ) ∞ X j =1 λ γ +2 j b j b j +1 − c ( ν + µ ) X j ≥ λ γ +2 j a j b j − ν k a k γ +1 − µ k b k γ +1 + c ∞ X j =1 λ γ + θj a j + c ∞ X j =1 λ γ + θj b j . (4.35)In view of the inequalities in Lemma 3.1 (i) and (iii) and (4.35), we have ddt L ( t ) ≥ (cid:18) − ν − ν (1 + λ ) λ − γ − − c ( ν + µ ) (cid:19) k a k γ +1 + (cid:18) − µ − µ (1 + λ ) λ − γ − − c ( ν + µ ) (cid:19) k b k γ +1 + c c k a k γ +1 + c c k b k γ +1 ≥ − M (cid:0) k a k γ +1 + k b k γ +1 (cid:1) + 12 c c (cid:0) k a k γ +1 + k b k γ +1 (cid:1) = (cid:0) k a k γ +1 + k b k γ +1 (cid:1) (cid:18) c c (cid:0) k a k γ +1 + k b k γ +1 (cid:1) − M (cid:19) (4.36)where we denote M := 2( ν + µ ) + ( ν + µ )(1 + λ ) λ − γ − + c ( ν + µ ) .In the following, we will show that for an appropriate constant M > theassumption k a (0) k γ + k b (0) k γ > M can close the argument. Indeed, we define M := 4 M c c (2 + 2 λ − γ − ) > M c c . (4.37)Thus, it follows from the assumption k a (0) k γ + k b (0) k γ > M that k a (0) k γ +1 + k b (0) k γ +1 ≥ k a (0) k γ + k b (0) k γ > M LOW-UP OF DYADIC MHD MODELS 15 and hence by (4.37) we have c c (cid:0) k a (0) k γ +1 + k b (0) k γ +1 (cid:1) − M > c c M − M ≥ M > . Therefore, (4.36) implies that ddt L ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 > , and hence, there exists a small time T > such that L ( t ) > L (0) , ∀ t ∈ (0 , T ] . (4.38)We are left to show that L satisfies a Riccati type of inequality. Based on (4.36),we just need to show that c c (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) − M ≥ (4.39)for t on a small time interval, which is expected due to the largeness assumptionon the initial data. In fact, from the definition of L ( t ) in (3.9), it is obviously truefor any t ≥ k a ( t ) k γ + k b ( t ) k γ ≤ L ( t ) ≤ (cid:0) c + c ) λ − γ − + c (cid:1) (cid:0) k a ( t ) k γ + k b ( t ) k γ (cid:1) (4.40)in view of the inequalities of Lemma 3.1 (iii) with γ + 1 replaced by γ . Thus, (4.38)and (4.40) imply L ( t ) ≥ L (0) ≥ k a (0) k γ + k b (0) k γ > M , ∀ t ∈ [0 , T ] . (4.41)The right hand side of (4.40) also indicates for any t ≥ L ( t ) ≤ (cid:0) λ − γ − (cid:1) (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) . (4.42)We conclude from (4.41) and (4.42) k a ( t ) k γ +1 + k b ( t ) k γ +1 ≥ L ( t )2 + 2 λ − γ − > M λ − γ − , t ∈ [0 , T ] , and hence the definition of M in (4.37) implies (4.39). As a consequence, we haveon [0 , T ] (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) (cid:18) c c (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) − M (cid:19) ≥ c c (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) . (4.43)It follows from (4.36), (4.43) and (4.42) that ddt L ( t ) ≥ c c (cid:0) k a ( t ) k γ +1 + k b ( t ) k γ +1 (cid:1) ≥ c c (2 + 2 λ − γ − ) − L ( t ) , t ∈ [0 , T ] . (4.44)In the end, we point out that since L ( T ) ≥ L (0) > M , we can start at time T andrepeat the process above iteratively to show that the Riccati type inequality (4.44)holds for all t ≥ . It indicates that L ( t ) blows up in finite time. (cid:3) LOW-UP OF DYADIC MHD MODELS 16 Blow-up of positive solutions of dyadic Hall-MHD
In this section, we prove the blow up of positive solution to the Hall MHD dyadicmodel (1.7) with d i > . The strategy of the proof is similar to that of Theorem1.1 for the MHD dyadic model. With the preparation of Lemma 3.3, in order toprove Theorem 1.2, it is sufficient to show the following lemma. Lemma 5.1.
Consider system (1.7) with d i > . Let θ > γ and < γ ≪ .Assume k a (0) k γ + k b (0) k γ > M for a certain constant M > . The function L ( t ) defined in (3.10) for positive solution ( a ( t ) , b ( t )) of (1.7) is a Lyapunov functionand it blows up in finite time. Proof:
The main step is to establish a Riccati type inequality for L . To doso, direct computation based on (1.7) with d i > ( d i = 1 is taken to reduce thenumber of parameters) gives us ddt (cid:16) λ γj a j a j +1 (cid:17) = − ν (1 + λ ) λ γ +2 j a j a j +1 + λ γ + θj a j + λ γ + θj a j b j + λ θj − λ γj a j − a j +1 + λ θj − λ γj b j − a j +1 − λ γ + θj b j a j +1 b j +1 − λ γj λ θj +1 a j b j +1 b j +2 − λ γ + θj a j a j +1 − λ γj λ θj +1 a j a j +1 a j +2 , (5.45) ddt (cid:16) λ γj b j b j +1 (cid:17) = − µ (1 + λ ) λ γ +2 j b j b j +1 + λ γ + θ +1 j b j + λ γ + θj a j b j +1 + λ γj λ θj +1 b j a j +1 b j +2 + λ θ +1 j − λ γj b j − b j +1 − λ γ + θj b j a j +1 b j +1 − λ γj λ θj +1 b j b j +1 a j +2 − λ γ + θ +1 j b j b j +1 − λ γj λ θ +1 j +1 b j b j +1 b j +2 , (5.46) ddt (cid:0) k a ( t ) k γ + k b ( t ) k γ (cid:1) = − ν k a ( t ) k γ +1 − µ k b ( t ) k γ +1 + 2( λ γ − ∞ X j =1 λ γ + θj a j a j +1 + 2( λ γ − ∞ X j =1 λ γ + θj b j a j +1 + 2( λ γ − ∞ X j =1 λ γ + θ +1 j b j b j +1 . (5.47)The negative terms on the right hand side of (5.45)-(5.47) are estimated below, byYoung’s inequality λ γ + θj a j a j +1 = λ − (2 γ + θ ) (cid:16) λ (2 γ + θ ) j a j a j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +1 a j +1 (cid:17) ≤ λ − (2 γ + θ ) λ γ + θj +1 a j +1 + 12 λ − (2 γ + θ ) λ γ + θj a j a j +1 ; (5.48) λ γ + θj b j a j +1 b j +1 = λ − (2 γ + θ ) (cid:16) λ (2 γ + θ ) j b j a j +1 (cid:17) (cid:16) λ (2 γ + θ ) j +1 a j +1 b j +1 (cid:17) ≤ λ − (2 γ + θ ) λ γ + θj b j a j +1 + 12 λ − (2 γ + θ ) λ γ + θj +1 a j +1 b j +1 ; (5.49) LOW-UP OF DYADIC MHD MODELS 17 λ γj λ θj +1 a j a j +1 a j +2 = λ γj λ θj +1 (cid:16) a j a j +1 (cid:17) (cid:16) a j +1 a j +2 (cid:17) (cid:16) a j +2 (cid:17) ≤ λ θ λ γ + θj a j a j +1 + 14 λ − γ λ γ + θj +1 a j +1 a j +2 + 14 λ − γ − θ λ γ + θj +2 a j +2 ; (5.50) λ γj λ θj +1 a j b j +1 b j +2 = λ − (2 γ +1) λ − j (cid:16) λ (2 γ + θ ) j a j (cid:17) (cid:16) λ (2 γ + θ +1) j +1 b j +1 (cid:17) (cid:16) λ (2 γ + θ +1) j +2 b j +2 (cid:17) ≤ λ − (2 γ + ) λ γ + θj a j + 13 λ − (2 γ + ) λ γ + θ +1 j +1 b j +1 + 13 λ − (2 γ + ) λ γ + θ +1 j +2 b j +2 ; (5.51) λ γj λ θj +1 b j b j +1 a j +2 = λ γj λ θj +1 (cid:16) b j b j +1 (cid:17) (cid:16) b j +1 (cid:17) ( a j +2 ) ≤ λ γj λ θj +1 b j b j +1 + 16 λ γj λ θj +1 b j +1 + 13 λ γj λ θj +1 a j +2 ≤ λ θ − λ γ + θ +1 j b j b j +1 + 16 λ − γ − λ γ + θ +1 j +1 b j +1 + 13 λ − γ − θ λ γ + θj +2 a j +2 ; (5.52) λ γ + θ +1 j b j b j +1 = λ γ + θ +1 j (cid:16) b j b j +1 (cid:17) (cid:16) b j +1 (cid:17) ≤ λ γ + θ +1 j b j b j +1 + 12 λ − (2 γ + θ +1) λ γ + θ +1 j +1 b j +1 ; (5.53) λ γj λ θ +1 j +1 b j b j +1 b j +2 = λ γj λ θ +1 j +1 (cid:16) b j b j +1 (cid:17) (cid:16) b j +1 b j +2 (cid:17) (cid:16) b j +2 (cid:17) ≤ λ θ +1 λ γ + θ +1 j b j b j +1 + 14 λ − γ λ γ + θ +1 j +1 b j +1 b j +2 + 14 λ − γ − θ − λ γ + θ +1 j +2 b j +2 . (5.54) LOW-UP OF DYADIC MHD MODELS 18
Applying (5.48)-(5.51) to (5.45) yields ddt c ∞ X j =1 λ γj a j a j +1 ≥ − c ν (1 + λ ) ∞ X j =1 λ γ +2 j a j a j +1 + c (cid:18) − λ − (2 γ + ) − λ − (2 γ + θ ) − λ − γ − θ (cid:19) ∞ X j =1 λ γ + θj a j + c (cid:18) − λ − (2 γ + θ ) (cid:19) ∞ X j =1 λ γ + θj a j b j − c λ − (2 γ + ) ∞ X j =1 λ γ + θ +1 j b j − c λ − (2 γ + θ ) ∞ X j =1 λ γ + θj b j a j +1 − c (cid:18) λ − (2 γ + θ ) + 12 λ θ + 14 λ − γ (cid:19) ∞ X j =1 λ γ + θj a j a j +1 . (5.55)While (5.49) and (5.52)-(5.54) applied to (5.46) gives ddt c ∞ X j =1 λ γj b j b j +1 ≥ − c µ (1 + λ ) ∞ X j =1 λ γ +2 j b j b j +1 − c λ − γ − θ ∞ X j =1 λ γ + θj a j − c λ − (2 γ + θ ) ∞ X j =1 λ γ + θj a j b j + c (cid:18) − λ − γ − − λ − γ − θ − − λ − γ − θ − (cid:19) ∞ X j =1 λ γ + θ +1 j b j − c λ − (2 γ + θ ) ∞ X j =1 λ γ + θj b j a j +1 − c (cid:18) λ θ − + 12 + 12 λ θ +1 + 14 λ − γ (cid:19) ∞ X j =1 λ γ + θ +1 j b j b j +1 . (5.56)In order to have the negative terms in (5.55)-(5.56) and (5.47) absorbed by thepositive terms, we claim there exists a constant c > such that c (cid:18) − λ − (2 γ + ) − λ − (2 γ + θ ) − λ − γ − θ (cid:19) − c λ − γ − θ ≥ c , (5.57) c (cid:18) − λ − γ − − λ − γ − θ − − λ − γ − θ − (cid:19) − c λ − γ − ≥ c , (5.58) c (cid:18) − λ − (2 γ + θ ) (cid:19) − c λ − (2 γ + θ ) ≥ , (5.59) LOW-UP OF DYADIC MHD MODELS 19 λ γ − − c (cid:18) λ − (2 γ + θ ) + 12 λ θ + 14 λ − γ (cid:19) ≥ , (5.60) λ γ − − c λ − (2 γ + θ ) − c λ − (2 γ + θ ) ≥ , (5.61) λ γ − − c (cid:18) λ θ − + 12 + 12 λ θ +1 + 14 λ − γ (cid:19) ≥ . (5.62)As a matter of fact, we can choose c = c and < c ≪ such that c ≤ λ γ − λ θ − + 2 + 2 λ θ +1 + λ − γ . (5.63)One can check conditions (5.59)-(5.62) are satisfied. Consequently, for λ ≥ , thereexists a constant c > such that (5.57) and (5.58) are also satisfied.In view of (3.10), adding (5.47) and (5.55)-(5.56) leads to ddt L ( t ) ≥ − c ν (1 + λ ) ∞ X j =1 λ γ +2 j a j a j +1 − c µ (1 + λ ) ∞ X j =1 λ γ +2 j b j b j +1 − ν k a k γ +1 − µ k b k γ +1 + c ∞ X j =1 λ γ + θj a j + c ∞ X j =1 λ γ + θ +1 j b j . (5.64)Applying the inequalities of Lemma 3.1 to (5.64), we obtain ddt L ( t ) ≥ (cid:0) − ν − c ν (1 + λ ) λ − γ − (cid:1) k a k γ +1 + (cid:0) − µ − c µ (1 + λ ) λ − γ − (cid:1) k b k γ +1 + c c k a k γ +1 + c c k b k γ +1 ≥ − M (cid:0) k a k γ +1 + k b k γ +1 (cid:1) + 12 c c (cid:0) k a k γ +1 + k b k γ +1 (cid:1) = (cid:0) k a k γ +1 + k b k γ +1 (cid:1) (cid:18) c c (cid:0) k a k γ +1 + k b k γ +1 (cid:1) − M (cid:19) (5.65)with M := 2( ν + µ ) + ( c ν + c µ )(1 + λ ) λ − γ − . Define M := 4 M c c (1 + ( c + c ) λ − γ − ) > M c c . With such M and the estimate (5.65), an analogous analysis as the last part ofthe proof of Lemma 4.1 can be used to justify the statement of the current lemma. (cid:3) Acknowledgement
The author is sincerely grateful to Professor Susan Friedlander for sharing somereferences on dyadic MHD models from the physics community, and for many in-sightful conversations with her.
LOW-UP OF DYADIC MHD MODELS 20
References [1] D. Barbato, F. Flandoli, and F. Morandin.
Energy dissipation and self-similar solutions foran unforced inviscid dyadic model . Trans. Amer. Math. Soc., 363 (4): 1925–1946, 2011.[2] D. Barbato and F. Morandin.
Positive and non-positive solutions for an inviscid dyadicmodel: well-posedness and regularity . Nonlinear Differential Equations Appl., 20 (3): 1105–1123, 2013.[3] D. Barbato, F. Morandin, and M. Romito.
Smooth solutions for the dyadic model . Nonlin-earity, 24 (11): 3083–3097, 2011.[4] L. Biferale.
Shell models of energy cascade in turbulence . Annu. Rev. Fluid Mech., 35:441468, 2003.[5] D. Biskamp.
Cascade models for magnetohydrodynamic turbulence . Phys. Rev. E50: 2702–2711, 1994.[6] A. Cheskidov.
Blow-up in finite time for the dyadic model of the Navier-Stokes equations .Trans. Amer. Math. Soc., 360 (10): 5101-5120, 2008.[7] A. Cheskidov and M. Dai.
Kolmogorov’s dissipation number and the number of degrees offreedom for the 3D Navier-Stokes equations . Proceedings of the Royal Society of Edinburg,Section A, Vol. 149, Issue 2: 429–446, 2019.[8] A. Cheskidov and S. Friedlander.
The vanishing viscosity limit for a dyadic model . PhysicaD, 238:783–787, 2009.[9] A. Cheskidov, S. Friedlander, and N. Pavlović.
Inviscid dyadic model of turbulence: thefixed point and Onsager’s conjecture . J. Math. Phys., 48 (6): 065503, 16, 2007.[10] A. Cheskidov, S. Friedlander, and N. Pavlović.
An inviscid dyadic model of turbulence: theglobal attractor . Discrete Contin. Dyn. Syst., 26 (3): 781–794, 2010.[11] P. Constantin, B. Levant, and E.Titi.
Analytic study of the shell model of turbulence . PhysicaD: Nonlinear Phenomena, 219 (2): 120–141, 2006.[12] M. Dai.
Dyadic models with intermittency dependence for the Hall MHD . arXiv: 2006.15094,2020.[13] E. I. Dinaburg and Y. G. Sinai.
A quasi-linear approximation of three-dimensional Navier-Stokes system . Moscow Math. J., 1: 381–388, 2001.[14] S. Friedlander and N. Pavlović.
Blowup in a three-dimensional vector model for the Eulerequations . Comm. Pure Appl. Math., 57 (6): 705–725, 2004.[15] U. Frisch.
Turbulence: The Legacy of A. N. Kolmogrov . Cambridge University Press, Cam-bridge, 1995.[16] E. B. Gledzer.
System of hydrodynamic type admitting two quadratic integrals of motion .Soviet Phys. Dokl., 18: 216-217, 1973.[17] C. Gloaguen, J. Léorat, A. Pouquet and R. Grappin.
A scalar model for MHD turbulence .Physica D.: Nonlinear Phenomena, 17(2):154–182, 1985.[18] I. Jeong and D. Li.
A blow-up result for dyadic models of the Euler equations . Communica-tions in Mathematical Physics, 337:1027–1034, 2015.[19] N. Katz and N. Pavlović.
Finite time blow-up for a dyadic model of the Euler equations .Trans. Amer. Math. Soc., 357 (2): 695–708, 2005.[20] A. Kiselev and A. Zlatoš.
On discrete models of the Euler equation . Int. Math. Res. Not.,38: 2315–2339, 2005.[21] V. S. L’vov, E. Podivilov, A. Pomyalov, I. Procaccia, and D. Vandembroucq.
Improved shellmodel of turbulence . Phys. Rev. E (3) 58: 1811–1822, 1998.[22] A. M. Obukhov.
Some general properties of equations describing the dynamics of the atmo-sphere . Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer. i Okeana, 7:695–704, 1971.[23] K. Ohkitani and M. Yamada.
Temporal intermittency in the energy cascade process andlocal Lyapunov analysis in fully-developed model of turbulence . Progr. Theoret. Phys., 81:329–341, 1989.[24] F. Plunian, R. Stepanov and P. Frick.
Shell models of magnetohydrodynamic turbulence .Physics Reports, vol. 523, 2013.[25] F. Waleffe.
On some dyadic models of the Euler equations . Proc. Amer. Math. Soc., 134(10): 2913–2922, 2006.
LOW-UP OF DYADIC MHD MODELS 21
Department of Mathematics, Statistics and Computer Science, University of Illi-nois at Chicago, Chicago, IL 60607, USA
Email address ::