aa r X i v : . [ m a t h . A P ] M a r ON THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION
DANIEL LEAR
Abstract.
Motivated by an equation arising in magnetohydrodynamics, we address the well-posednesstheroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which thedivergence-free drift velocity is one derivative more singular that the active scalar. In [14], the authors provethat the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally wellposed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stablefor periodic perturbations with initial data localized in frequency straight lines crossing the origin. For suchwell-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces andthe global existence holds under a size condition over the H / + ( T ) norm of the perturbation. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 The perturbated system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63 Local existence for frequency-localized initial data . . . . . . . . . . . . . . . . . . . . .
104 Global existence in H / + ( T ) for frequency-localized initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
The geodynamo is the process by which the Earth’s magnetic field is created and sustained by the motionof the fluid core, which is composed of a rapidly rotating, density stratified, electrically conducting fluid. Theconvective processes in the core that produce the velocity fields required for dynamo action are a combinationof thermal and compositional convection. The full dynamo problem requires the examination of the full 3Dpartial differential equations governing convective, incompressible magnetohydrodyamics (MHD).It is therefore reasonable to attempt to gain some insight into the geodynamo by considering a reductionof the full MHD equations to a system that is more tractable, but one that retains many of the essentialfeatures relevant to the physics of the Earth’s core.Recently, Moffatt and Loper [22], [23] proposed the magneto-geostrophic equation (MG) as a model forthe geodynamo which is a reduction of the full MHD system. The physical postulates of this model are thefollowing: slow cooling of the Earth leads to slow solidification of the liquid metal core onto the solid innercore and releases latent heat of solidification that drives compositional convection in the fluid core.1.1.
Governing equations:
We first present the full coupled three-dimensional MHD equations for theevolution of the velocity vector U ( x , t ), the magnetic field vector B ( x , t ) and the buoyancy field Θ( x , t ) inthe Boussinesq approximation and written in the frame of reference rotating with angular velocity Ω. Forsimplicity, we have assumed that the axis of rotation and the gravity g are aligned in the direction of e .Following the notation of Moffatt and Loper [23] we obtain the dimensionless equations N [ R ( ∂ t U + U · ∇ U ) + e × U ] = −∇ P + ( e · ∇ ) b + R m b · ∇ b + N Θ e + ǫ ν ∆ U ,R m [ ∂ t b + U · ∇ b − b · ∇ U ] = ( e · ∇ ) U + ∆ b ,∂ t Θ + U · ∇ Θ = ǫ κ ∆Θ , ∇ · U = 0 , ∇ · b = 0 , (1)where P is the sum of the fluid and magnetic pressures, ǫ ν is the (non-dimensional) kinematic viscosity and ǫ κ is the (non-dimensional) thermal diffusivity. Here ( e , e , e ) denote the Cartesian unit vectors. Date : March 7, 2019.
Following [22], we have assumed in (1) that the magnetic field in the core is of the form B ( x , t ) = B + b ( x , t ) , where B results from dynamo action and can be considered as locally uniform and steady, and a perturbationfield b ( x , t ) induced by the flow U ( x , t ) across B . Our choice of B ≡ β e as the underlying magnetic fieldis consistent with the models where the magnetic field is believed to be predominantly toroidal due to thestrong influence of differential rotation [23].The dimensionless parameters in (1) are the followings: N = 2Ω µ ηρ/β inverse of the Elsasser number ,R o = V / L Ω Rossby number ,R m = V L/η magnetic Reynolds number ,ǫ ν = νηµ ρ/β L inverse square of the Hartman number ,ǫ κ = κ/LV inverse of the Peclet number . Here ν is the kinematic viscosity, η is the magnetic diffusivity of the fluid, κ is the molecular diffusivity of thecompositional variation that creates an ambient density ρ and µ = 4 π × − NA − . We adopt as velocityscale V = Θ g/
2Ω where Θ is the typical amplitude of Θ, and that the length-scale of these variations is L .The orders of magnitude of the nondimensional parameters are motivated by the physical postulates ofthe Moffatt and Loper model. For the regions in the Earth’s fluid core modeled in (1), it is argued in [23]that the parameters have the following orders of magnitude: N ≈ , R o ≈ − , R m ≈ , ǫ ν ≈ − , ǫ κ ≈ − . The values of ν and κ are speculative, but likely to be extremely small. For a detailed discussion of plausibleranges of the physical parameters that are appropiate for the geodynamo, we refer the reader to [16].According to Moffat and Loper, the magnetic Reynolds number is relatively small. Then, their modelneglects the terms multiplied by R o and R m in comparison with the remaining terms. However, we will forthe moment retain the viscous and diffusive terms since it involve the highest derivatives.For the reasons given above, we now drop in (1) the terms involving the Rossby number R o and themagnetic Reynolds number R m . Then, we obtain the following reduced system: N [ e × U ] = −∇ P + ( e · ∇ ) b + N Θ e + ǫ ν ∆ U , e · ∇ ) U + ∆ b ,∂ t Θ + U · ∇ Θ = ǫ κ ∆Θ , ∇ · U = 0 , ∇ · b = 0 . (2)Essentially this means that the evolution equations for the coupled velocity U and magnetic field b take asimplified “quasi-static” form. This system encodes the vestiges of the physics in the problem, namely theCoriolis force, the Lorentz force and gravity.The behavior of the model is dramatically different when the parameters ǫ ν and ǫ κ are present or absent.Since both parameters multiply a Laplacian term, their presence is smoothing. In the present paper we focusour attention in the inviscid case ( ǫ ν = 0). The mathematical properties of the model under the presence ofviscosity have been addressed in a recent sequence of different articles [10], [11] and [12].1.2. The MG equation:
A linear relationship can be established between the divergence-free vector fields U and b and the scalar Θ, wherein Θ will now be regarded as known, thanks to the reduced system: N [ e × U ] = −∇ P + ( e · ∇ ) b + N Θ e , e · ∇ ) U + ∆ b , (3)along with the incompressibility condition ∇ · U = 0 , ∇ · b = 0 . N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 3
We note that the ratio of the Coriolis to Lorentz forces in their model is of order 1, so for notationalsimplicity we have set this parameter, denoted by N equal to 1. Following [13, p. 297], manipulations ofthe linear system (3) gives, in component form U = − D − ( ∂ P + Γ ∂ P ) ,U = D − ( ∂ P − Γ ∂ P ) ,∂ U = D − Γ∆ H P,∂ Θ = (cid:0) Γ ∆ H D − + ∂ (cid:1) P, (4)where the operators Γ , D and ∆ H are defined asΓ := − ( − ∆) − ∂ , D := 1 + Γ , ∆ H := ∂ + ∂ . Although the physically relevant boundary for a model of the Earth’s fluid core is a spherical annulus, formathematical tractability we considered the system on the domain x ∈ T × R . This can be seen as a firststep before considering the case T × I with appropiate boundary conditions in the vertical variable.In order to uniquely determine U and Θ form (4), we restrict the system to the function spaces of zerovertical mean, i.e. ´ R U dx = ´ R Θ dx = 0. In fact, without such a restriction the system is not well defined.We can integrate the last equation of (4) and use the zero vertical mean assuption to obtain thatΘ = A [ P ] , where the operator A is formally defined as A := ∂ − (cid:0) Γ ∆ H D − + ∂ (cid:1) in the physical space. On one hand,we now use (4) to represent U , U and U in terms of Θ: U = − D − ( ∂ + Γ ∂ ) (cid:0) A − [Θ] (cid:1) ≡ M [Θ] ,U = D − ( ∂ − Γ ∂ ) (cid:0) A − [Θ] (cid:1) ≡ M [Θ] ,U = D − Γ∆ H (cid:0) D − Γ∆ H + ∂ (cid:1) − [Θ] ≡ M [Θ] . (5) Remark:
A precise expression of the operator M will be given as a Fourier multiplier operator in Section 2.On the other hand, the magnetic vector field b is computed from the scalar Θ thanks to (3) via the operator b j = ( − ∆) − ∂ M j [Θ] , for j ∈ { , , } . The sole remaining nonlinearity in the system comes from the coupling of (3) and the evolution equationfor the scalar bouyancy Θ. The active scalar equation for Θ that contains the non-linear process in Moffatt’smodel is precisely: (cid:26) ∂ t Θ + U · ∇ Θ = ǫ κ ∆Θ , ∇ · U = 0 , (6)where the divergence-free velocity U is explicitly obtained from the bouyancy as U = M [Θ] where M is thenon-local differential operator of order 1 defined in (5). We describe below the precise form of that operator. Remark:
As we said, we consider for simplicity the domain T × R . Note that, without loss of generalitywe may assume that ´ T × R Θ( x , t ) d x = 0 for all t ≥
0, since the mean of Θ is conserved by the flow.In the following, we refer to the evolution equation (6) with singular drift velocity U given by (5) asthe magneto-geostrophic equation (MG). In addition, we will distinguish between diffusive ( ǫ κ >
0) and non-diffusive case ( ǫ κ = 0). In the Earth’s fluid core the value of the diffusivity ǫ κ is very small. Hence it isrelevant to address both the diffusive evolution, and the non-diffusive version where ǫ κ = 0 . The aim of the present paper is to show that the Cauchy problem for the non-diffusive
MG equation iswell-posed with respect to some periodic perturbations around a specific steady profile, in the topology of acertain Sobolev space. In the next section, we state the main result of this paper at a descriptive level.1.3.
Diffusive vs. non-diffusive MG equation:
In order to study this dichotomy, we recall the following:In the theory of differential equations, it is classical to call a Cauchy problem well-posed , in the sense ofHadamard, if given any initial data in a functional space X , the problem has a unique solution in L ∞ (0 , T ; X ),with T depending only on the X -norm of the initial data, and moreover the solution map Y L ∞ (0 , T ; X )satisfies strong continuity properties, e.g. it is uniformly continuous, Lipschitz, or even C ∞ smooth, for asufficiently nice space Y ⊂ X . If one of these properties fail, the Cauchy problem is called ill-posed . DANIEL LEAR
Considering this, both systems have contrasting properties: • Diffusive MG equation: For ǫ κ > C ∞ smooth for positive times, as it is proved in the papers [13] and [15]. • Non-diffusive MG equation: For ǫ κ = 0, in [14] the authors prove that the equation is ill-posed inthe sense of Hadamard in Sobolev spaces, but locally well-posed in spaces of analytic functions.More specifically, we mention that for analytic initial data, the non-diffusive MG equation is indeed locallywell-posed in the class of real-analytic functions in the spirit of a Cauchy-Kowalewskaya result, since eachterm in the equation loses at most one derivative.Moreover, in the same article the authors prove that the solution map associated to the Cauchy problemis not Lipschitz continuous with respect to perturbations in the initial data around a specific steady profileΘ ( x ) := sin( m x ) for some integer m ≥
1, in the topology of a certain Sobolev space.The proof consists of a linear and a nonlinear step. After linearizing the problem around Θ , the authorsemploy techniques from continued fractions in order to construct an unstable eigenvalue for the linearizedoperator. Once these eigenvalues are exhibited, one may use a fairly robust argument to show that thissevere linear ill-posedness implies the Lipschitz ill-posedness for the nonlinear problem.The use of continued fractions in a fluid stability problem was introduced in [21] for the Navier-Stokesequations and later adapted for the Euler equations in [9].Hence, without the Laplacian to control the unbounded operator M the situation is dramatically differentfrom the diffusive case ǫ κ >
0. For the above, the problem of the fractionally diffusive
MG equation arisenaturally. This is, one can replace the Laplacian by nonlocal operators, such as − ( − ∆) γ for γ ∈ (0 , γ ∈ (1 / ,
1) the equation is locally well-posed, while it is Hadamard Lipschitz ill-posed for γ ∈ (0 , / γ = 1 / M can be explored as in [8] to obtain animprovement in the regularity of the solutions when the initial data is supported on a plane in the Fourierspace. For such well-prepared initial data the local existence and uniqueness of solutions can be obtainedfor all values γ ∈ (0 , γ ∈ (1 / , Singular active scalar.
One may view the MG equation as an example of a singular active scalar sincethe drift velocity is given in terms of the advected scalar by a constitutive law which is losing derivatives.Active scalars appear in many problems coming from fluid mechanics. It consists of solving the Cauchyproblem for the transport equation: (cid:26) ∂ t Θ + U · ∇ Θ = − ǫ κ ( − ∆) γ Θ , ∇ · U = 0 , (7)where the vector field U is related to Θ by some operator. We remark that the MG equation fall into ahierarchy of active scalar equations arising in fluid dynamics in terms of the nature of the operator thatproduces the drift velocity from the scalar field: Hierarchy of active scalar equations: i) Inviscid MG equation ( ε ν = 0): U = M [Θ] Singular order 1 • ǫ κ = 0: Hadamard ill-posed. • ǫ κ >
0: Critical case, globally well-posed.ii) SGQ equation (see [4] and [2], [5], [18]): U = ∇ ⊥ ( − ∆) − / Θ Singular order 0 • ǫ κ = 0: Open. • ǫ κ >
0: Critical case, globally well-posed.iii) Burgers equation (see [19]): U = Θ Order 0 • ǫ κ = 0: Blow-up. • ǫ κ >
0: Critical case, globally well-posed.iv) 2D Euler equation in vorticity form: U = ∇ ⊥ ( − ∆) − Θ Smoothing degree 1
Globally well-posed.
N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 5
We emphasize that the mechanism producing ill-posedness is not merely the order one derivative loss inthe map Θ U . Rather, it is the combination of the derivative loss with the anisotropy of the symbol M and the fact that this symbol is even. We note that the even nature of the symbol of M plays a centralrole in the proof of non-uniqueness for L ∞ -weak solutions to the non-diffusive MG equation proved in [25],via methods from convex integration. In contrast, an example of an active scalar equation where the mapΘ U is unbounded, but given by an odd Fourier multiplier, is the generalized SQG equation where U = ∇ ⊥ ( − ∆) − − γ Θ and 0 < γ ≤
1. This equation was recently shown in [3] to give a locally well-posedproblem in Sobolev spaces.1.4.1.
On the lack of well-posedness for the MG equation in Sobolev spaces.
Let us briefly discuss why theevenness of the operator M breaks the classical proof of local existence in Sobolev spaces for the ( ǫ κ = 0) non-diffusive MG equation. To see why one may not use the standard energy-approach to obtain local wellposedness, we point out that in the energy estimate for (7) there are only two terms which seem to preventclosing the estimate at the H s level: T bad = ˆ Λ s Θ Λ s U j ∂ j Θ = ˆ Λ s Θ Λ s M j [Θ] ∂ j Θ and T good = ˆ Λ s Θ U j ∂ j Λ s Θ , where we denoted Λ := ( − ∆) − / . Since ∇ · U = 0, upon integrating by parts we have T good = 0. On theother hand, the term T bad does not vanish in general. The only hope to treat the term T bad would be todiscover a commutator structure. However, since M is not anti-symmetric, i.e. even in Fourier space, wecannot write T bad = −T , where T = ˆ M j [Λ s Θ ∂ j Θ] Λ s Θ = T bad + ˆ [ M j , ∂ j Θ] Λ s Θ Λ s Θ = T bad + S . If you could do this, a suitable commutator estimate of Coifman-Meyer type would close the estimates atthe level of Sobolev spaces. Instead we have that T bad = T . This is the main reason why we are unable toclose estimates at the Sobolev level.1.5. Preliminares.
This section contains a few auxiliary results used in the paper. In particular, we recallthe, by now classical, product and commutator estimates, as well as the Sobolev embedding inequalities.Proofs of these results can be found for instance in [17],[26] and [27].
Lemma 1.1 ( Product estimate). If s > , then for all f, g ∈ H s ∩ L ∞ we have the estimate k Λ s ( f g ) k L . ( k f k L ∞ k Λ s g k L + k Λ s f k L k g k L ∞ ) . (8)In the case of a commutator we have the following estimate. Lemma 1.2 ( Commutator estimate).
Suppose that s > . Then for all f, g ∈ S we have the estimate k Λ s ( f g ) − f Λ s g k L . (cid:0) k∇ f k L ∞ k Λ s − g k L + k Λ s f k L p k g k L p ′ (cid:1) (9) where = p + p ′ and p ∈ (1 , ∞ ) . Moreover, the following Sobolev embeddings holds: • W s,p ( T d ) ⊂ L q ( T d ) continuosly if s < d/p and p ≤ q ≤ dp/ ( d − sp ). • W s,p ( T d ) ⊂ C k ( T d ) continuosly if s > k + d/p .1.6. Notation & Organization:
To avoid clutter in computations, function arguments (time and space)will be omitted whenever they are obvious from context. Finally, we use the notation f . g when thereexists a constant C > f ≤ Cg .In Section 2 we begin by setting up the perturbated problem around a specific steady state. Then, we statea less technical version of the main theorem and we give some of the ideas behind the proof. Here, we collectsome useful technical lemmas about the behaviour of c M on suitable subsets of the frequency domain. InSection 3 we embark on the proof of a local existence result for frequency-localized initial data following theideas of [8]. The core of the article is the proof of the main theorem in Section 4. We start by the a priori energy estimates given in Section 4.1. This is followed by an explanation of the decay given by the linearsemigroup in Section 4.2. Finally, in Section 4.3 we exploit a bootstrapping argument to prove our theorem. DANIEL LEAR The perturbated system
With all the above in mind, we seek to find a steady state around which the non-diffusive
MG equation iswell-posed. Following the same idea used initially in [8], we take advantage of the anisotropy of the symbols M = ( M , M , M ) given by (5) and observe an interesting phenomenon: when the initial perturbation islocalized in the frequency space, it is possible to prove a well-posedness result for the ensuing solution.If the frequency of the initial perturbation of the steady state lies on a suitable region of the Fourier space,then the operator M behaves like an order zero opeartor, and hence the corresponding velocity is as smoothas the advected scalar. This enables us to obtain a well-posedness result over the generic setting when noconditions of the Fourier spectrum of the initial perturbation are imposed.2.1. The steady states.
When studying fluid equations, it is often helpful to have a good understandingof the exact steady states of the system. The kinds of exact solutions we are interested are the simplestpossible steady state, namely U = 0 and Θ = Ω( x ) for some function Ω with ´ R Ω( x ) dx = 0.The basic problem is to consider Θ a given equilibrium state and to study the dynamics of solutionswhich are close to it in a suitable sense. Now, we write the scalar and the velocity asΘ( x , t ) = Ω( x ) + θ ( x , t ) ,U ( x , t ) = u ( x , t ) , and the pressure term is written in a more convenient way as P ( x , t ) = Ω(0) + ˆ x Ω( s ) ds + p ( x , t ) . Then, putting this ansatz in (4) we obtain u = − D − ( ∂ p + Γ ∂ p ) ,u = D − ( ∂ p − Γ ∂ p ) ,∂ u = D − Γ∆ H p,∂ θ = (cid:0) Γ ∆ H D − + ∂ (cid:1) p. (10)As before, in order to uniquely determine u and θ from (10), we restrict the system to the function spaceswhere u and θ have zero vertical mean. Hence, we can integrate the last equation of the previous systemand use the zero vertical mean assuption to obtain that θ ( x , t ) = A [ p ]( x , t ) . Remark:
If we impose that θ ( x , t ) and p ( x , t ) are periodic functions in the three variables x = ( x , x , x ),then A is invertible on the space of functions with zero x -mean and has an expression as a Fourier multiplier.For periodic perturbations in the three variables, the operator A is a Fourier multiplier with symbolˆ A ( k ) = 1 ik k | k | + k | k | + k , where the Fourier variable k ∈ Z ⋆ := Z \ { k = 0 } , by our vertical mean-free assumption. After that, wecan use (10) and (3) to represent u and b in terms of θ via u j = M j [ θ ] and b j = ( − ∆) − ∂ M j [ θ ] for j ∈ { , , } . (11)Note that the operators { M j } j =1 are Fourier multipliers with symbols given explicity for k ∈ Z ⋆ by c M ( k ) := k k | k | − k k k k | k | + k , c M ( k ) := − k k | k | − k k k | k | + k , c M ( k ) := k ( k + k ) k | k | + k . On { k = 0 } we let c M j ( k ) = 0, since for consistency of the model we have that θ and U have zero x -mean.It can be directly checked that k j · c M j ( k ) = 0 and hence the velocity field u given by (11) is divergence-free. N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 7
The Fourier multiplier operator.
We study the properties and behavior of the Fourier mulplier operator M obtained from (11), which relates u and θ . It is important to note that although the symbols c M j arezero-order homogenous under the isotropic scaling k → λ k , due to their anisotropy the symbols c M j arenot bounded functions of k . In fact, it may be shown that | c M ( k ) | . | k | and this bound is sharp. To seethis, note that whereas in the region of Fourier space where | k | ≤ max {| k | , | k |} the c M j are bounded bya constant, uniformly in | k | , this is not the case on the “curved” frequency regions where k = O (1) and k = O ( | k | r ), with 0 < r ≤ /
2. In such regions the symbols are unbounded, since as | k | → ∞ we have: | c M ( k , | k | r , | ≈ | k | r , | c M ( k , | k | r , | ≈ | k | , | c M ( k , | k | r , | ≈ | k | r . (12)Several important properties of the c M j ’s are immediately obvious:i) The functions are strongly anisotropic with respect to the dependence on the integers k , k , and k .This is a consequence of the interplay of the three physical forces governing this system: • Coriolis force, • Lorentz force, • Gravity.ii) Since the symbols c M j are even the operator M is not anti-symmetric.These properties of c M make the MG equation interesting and challenging mathematically, as well as havinga clear physical basis in its derivation from the MHD equations.Finally, after put our ansatz in (6), we arrive to the following system: ∂ t θ ( x , t ) + u ( x , t ) · ∇ θ ( x , t ) = − ǫ κ ∆ θ ( x , t ) − ǫ κ Ω ′′ ( x ) − u ( x , t )Ω ′ ( x ) , u ( x , t ) = M [ θ ]( x , t ) ,θ ( x ,
0) = θ ( x ) , (13)where our initial data θ has zero vertical mean. Remark:
Here x = ( x , x , x ) ∈ T × R , however θ ( x , t ) and u ( x , t ) are periodic in the three variables.For the case Ω ≡
0, the system (13) is again the one widely studied in [7, 8, 13, 14, 15]. The aim of thispaper is to show that the Cauchy problem for the non-diffusive
MG equation is well-posed with respect toperturbations around a specific steady profile Ω, in the topology of a certain Sobolev space.2.2.
The perturbated non-diffusive
MG equation.
We fix the perturbation θ ( x , t ) := Θ( x , t ) − Ω( x ).Therefore, we obtain the system: ∂ t θ ( x , t ) + u ( x , t ) · ∇ θ ( x , t ) = − u ( x , t )Ω ′ ( x ) , u ( x , t ) = M [ θ ]( x , t ) ,θ ( x ,
0) = θ ( x ) , (14)with x ∈ T and where our initial data θ has zero vertical mean.What is interesting about this equation is that M is a positive operator so we get a mild dissipationeffect. This structure will allow us to prove stability. So, just as for the fractional Laplacian, we define thesquare root of M via Fourier transform as follows: Definition 2.1.
The square root of M can be defined on functions f : T → R with zero vertical mean asa Fourier multiplier given by the formula: \ p M [ f ]( k ) := s k ( k + k ) k | k | + k ˆ f ( k ) k ∈ Z ⋆ . (15) Note that [ √ M ( k ) is not defined on k = 0 since for the self-consistency of the model, we only work withperiodic functions with zero vertical mean. DANIEL LEAR
The main theorem.
In this work, we are interested in the perturbative regime near the special steadystate Ω( x ) := x . The main achievement of the paper is a local existence result for periodic perturba-tions localized in a suitable section of the frequency space together with a global existence result under anadditional size condition over the H / ( T ) norm of the perturbation.To sum up, we want to consider solutions in x ∈ T × R and t ≥ x , t ) := Ω( x ) + θ ( x , t ) , for periodic perturbations θ ( x ,
0) := θ ( x ) ∈ H s ( T ) with zero vertical mean and frequency support in X ⊂ Z .Then, we prove: • Local well-posedness: If s > . • Global well-posedness: If s > and || θ || H / is small enough. • GWP & asymptotic stability: If s > and || θ || H / is small enough.A precise statement of our result is presented as Theorem 4.1, where we also illustrate its proof through abootstrap argument. Despite the apparent simplicity, understanding the stability of this flow is non-trivial.2.2.2. The ideas behind the proof:
In order to prove this, first we fix our attention in the study of the stabilityof the problem, when linearized it around a particular steady state Ω( x ) ≡ x . The main mechanism ofdecay can be seen from the linearized equation: (cid:26) ∂ t θ ( x , t ) = − M [ θ ]( x , t ) ,θ ( x ,
0) = θ ( x ) . As c M ( k ) is a positive operator for k ∈ Z ⋆ ≡ Z \ { k = 0 } there is a unique positive self-adjoint square rootoperator of c M ( k ) on Z ⋆ , which we define in (15). In consequence, the linearized equaiton clearly shows thedecay over time of θ ( x , t ), except for the zero mode in x . However, we do not have that problem becausefor self-consistency of the model we restrict to functions that have zero vertical mean.Hence, the main achievement of the paper is thus to control the nonlinearity, so that it would not destroythe decay provided by the linearized equation. Note that, over the curved frequency regions where k = O (1)and k = O ( | k | r ) with 0 < r ≤ /
2, we have that u ( x , t ) ≈ Λ θ ( x , t ) and M [ θ ]( x , t ) ≈ Λ r θ ( x , t ) with0 < r ≤ / θ ( x , t ) lies on a suitable section of the Fourierspace, then the operator M behaves like an order zero operator and hence the corresponding velocity u ( x , t )is as smooth as θ ( x , t ). This enables us to obtain a well-posedness results over the generic setting whenno conditions on the Fourier spectrum of the initial perturbation θ are imposed. To be more precise, weconsider an appropiate subset X ⊂ Z which we will define later, where we can obtain a local well-posednessresult for perturbations θ ( x ) such that supp( b θ ( k )) ⊂ X.Under this hypothesis over the initial perturbation, at least morally speaking, our perturbated systembehaves like an active scalar of order zero with a damping term: (cid:26) ∂ t θ ( x , t ) + u ( x , t ) · ∇ θ ( x , t ) = − θ ( x , t ) ,θ ( x ,
0) = θ ( x ) , with u ( x , t ) = M [ θ ]( x , t ). However, as lim α → Λ α θ ( x , t ) = − θ ( x , t ), the type of results obtained for thesupercritical diffusive MG equation in [8] are expected to have also in our setting.2.2.3. Well-prepared initial data θ . In this section we explore the observation cited above: if the frequencysupport of θ lies on a suitable subset of the frecuency space, then the operator M is mild when it acts on θ ,i.e. it behaves like an order zero operator, and hence the corresponding velocity u is as smooth as θ .This enables us to obtain a well-posedness result over the generic setting when no conditions on theFourier spectrum of the initial perturbation are imposed. For instance, the local existence and uniquenessof smooth solutions holds for the non-diffusive case, a setting in which we know that for generic initial datathe problem is ill-posed in Sobolev spaces.For the perturbated problem, we are working in the periodic setting T and the frequency space is Z .Then, we can define the frequency straight lines L( q ) crossing the origin as the set:L( q ) := Z ∩ { ( q k, q k, q k ) : k ∈ Z } for q := ( q , q , q ) ∈ Q \ { (0 , , } . N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 9
Now, we will said that q ∈ Q \ { (0 , , } is an admissible triple if there exists C > q ∈ K C ,where K C is the rational cone defined byK C := (cid:8) q ∈ Q \ { (0 , , } : | q | , | q | ≤ C | q | (cid:9) . (16)The next lemma states that M behaves like a zero order operator when it acts on functions with frequencysupport in L( q ) with q ∈ K C for some C >
0. In the rest, we shall make key use of the next properties of M .In the rest of the paper, fixed C > C ∈ {{ L( q ) } : q ∈ K C } . This is, in the following forX C we will understand one of the previously defined frequency straight lines Lemma 2.2.
Let C > . For every smooth periodic function f : T → R with zero vertical mean andfrequency support in X C , there exists a universal constant m ⋆ = m ⋆ ( C ) > such that: (cid:12)(cid:12) \ M j [ f ]( k ) (cid:12)(cid:12) = (cid:12)(cid:12) c M j ( k ) ˆ f ( k ) (cid:12)(cid:12) ≤ m ⋆ (cid:12)(cid:12) ˆ f ( k ) (cid:12)(cid:12) with j ∈ { , , } for all k ∈ Z . Moreover, the constant m ⋆ blow-up as C tends to infinity.Proof. It is clear that the bound has to be proven only for k ∈ Z ⋆ , since otherwise we have that ˆ f ( k ) = 0and the statement holds trivially. Note that k ∈ X C implies that k = k · q with q ∈ K C and some k ∈ Z .We now consider each of the cases j ∈ { , , } . • For j = 1, a short algebraic computation gives (cid:12)(cid:12) c M ( k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k k | k | − k k k k | k | + k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) ( C + 2 C ) + C (cid:3) q q | q | + q . • Similarly to the previous one, it follows for j = 2 and j = 3 that: (cid:12)(cid:12) c M ( k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k k | k | + k k k | k | + k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) ( C + 2 C ) + C (cid:3) q q | q | + q and (cid:12)(cid:12) c M ( k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k ( k + k ) k | k | + k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + C ) q q | q | + q . It follows that | c M j ( k ) | ≤ m j ( C ) for j ∈ { , , } and taking m ⋆ := max { m , m , m } concludes the proof. (cid:3) Thanks to (16), it is simple to obtain an upper and lower bound for c M ( k ) in X C . The lower bound willplay a key role in the proof of the local and global existence result. The next lemma gives us this bound. Lemma 2.3.
Let C > . For every smooth periodic function f : T → R with zero vertical mean andfrequency support in X C , there exists a universal constant m ⋆ = m ⋆ ( C ) > such that: m ⋆ (cid:12)(cid:12) ˆ f ( k ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) \ M [ f ]( k ) (cid:12)(cid:12) for all k ∈ Z . Moreover, the constant m ⋆ goes to zero as C tends to infinity.Proof. It is clear that the bound has to be proven only for k ∈ Z ⋆ , since otherwise we have that ˆ f ( k ) = 0and the statement holds trivially. As M is a Fourier multiplier operator, we have that: | ˆ f ( k ) | = | [ M − ( k ) \ M [ f ]( k ) | ≤ || [ M − || L ∞ (X C ) | \ M [ f ]( k ) | . Morover, for k ∈ X C we have the bound [ M − ( k ) = k | k | + k k ( k + k ) ≤ (cid:2) ( C + 2 C ) + 1 (cid:3) q q ( q + q )from which it follows that | ˆ f ( k ) | ≤ m ⋆ | \ M [ f ]( k ) | for a suitable constant m ⋆ . (cid:3) As a consequence of the previous lemmas, under the same hypothesis as before we have that the Fourieroperator M is equivalent to the identity operator in L ( T ). More specifically, there exists a pair of realnumbers 0 < m ⋆ ≤ m ⋆ such that: m ⋆ || f || L ( T ) ≤ || M [ f ] || L ( T ) ≤ m ⋆ || f || L ( T ) . Corollary 2.4.
Let C > . For every smooth periodic function f : T → R with zero vertical mean andfrequency support in X C , there exists positive constants m ⋆ and m ⋆ such that: m ⋆ (cid:12)(cid:12) ˆ f ( k ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) \ M [ f ]( k ) (cid:12)(cid:12) ≤ m ⋆ (cid:12)(cid:12) ˆ f ( k ) (cid:12)(cid:12) for all k ∈ Z . The key point of the result of well-posedness is the fact that we only work with frequency localized initialperturbations. As we will see later in the proof, to prove that the perturbation does not leave the region ofthe frequency space where the operator M behaves like a zero order operator, the sets closed under additionwill play a crucial role.2.2.4. Closed sets under addition.
A set X is closed under addition + : X × X → X if for all a, b ∈ X we havethat a + b ∈ X. In other words, performing the binary operation on any two elements of the set always givesyou back something that is also in the set.
Lemma 2.5.
Fixed C ∈ Q and X C . For every pair of smooth periodic function f, g : T → R with frequencysupport in X C we have that: • supp (cid:16) c f g (cid:17) ⊂ X C . • supp (cid:16) [ f ± g (cid:17) ⊂ X C . • supp (cid:16) \ M j [ f ] (cid:17) ⊂ X C for all j ∈ { , , } .Proof. The proof is an immediate consequence of the properties of the Fourier transform: • Clearly supp( c f g ) = supp( ˆ f ∗ ˆ g ) ⊂ supp( ˆ f ) + supp(ˆ g ) ⊂ X C , since X C is closed under addition. • Note that supp( [ f ± g ) = supp( ˆ f ± ˆ g ) ⊂ supp( ˆ f ) ∪ supp(ˆ g ) ⊂ X C . • As M is a Fourier multiplier, we have: supp( \ M j [ f ]) = supp( c M j ˆ f ) ⊂ supp( c M j ) ∩ supp( ˆ f ) ⊂ X C . (cid:3) Local existence for frequency-localized initial data
The main result of this section is:
Theorem 3.1.
Fixed C ∈ Q and X C . Assume that θ ∈ H s ( T ) with s > / has zero vertical mean andsatisfies that supp ( b θ ) ⊂ X C . Then, there exists a time T > and a unique smooth solution θ ∈ L ∞ (0 , T ; H s ( T )) of the Cauchy problem (14) such that supp ( b θ ( t )) ⊂ X C for all t ∈ [0 , T ) . Before that, the goal is to prove the existence of smooth solutions to the scalar linear equation: (cid:26) ∂ t θ ( x , t ) + v ( x , t ) · ∇ θ ( x , t ) = − M [ θ ]( x , t ) θ ( x ,
0) = θ ( x ) (17)where the initial datum θ and the given divergence-free drift velocity field v satisfies that: • supp( b θ ) ⊂ X C . • supp( b v ( t )) ⊂ X C for all t ∈ [0 , T ) for a positive time T .The main result is: Theorem 3.2.
Let s > / . Given θ ∈ H s ( T ) and a divergence-free vector field v ∈ L ∞ (0 , T ; H s ( T )) satisfying the above conditions. Then, there exists a unique smooth solution of (17) such that: θ ∈ L ∞ (0 , T ; H s ( T )) . (18) Moreover, we have that supp (ˆ θ ( t )) ⊂ X C for all t ∈ [0 , T ) . Proof of Theorem 3.2.
Following the arguments of [8], we regularize (17) with hyper-dissipation as: (cid:26) ∂ t θ ǫ ( x , t ) + v ( x , t ) · ∇ θ ǫ ( x , t ) − ǫ ∆ θ ǫ ( x , t ) = − M [ θ ǫ ]( x , t ) θ ǫ ( x ,
0) = θ ( x ) (19)for ǫ ∈ (0 ,
1] and finally we pass to the limit ǫ → N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 11
On one hand, since v is smooth and divergence-free, it follows from the De Giorgi techniques (see [8] or[24]) that there exists a unique global smooth solution θ ǫ of (19) with θ ǫ ∈ L ∞ (0 , T ; H s ( T )) ∩ ǫL (0 , T ; H s +1 ( T )) . On the other hand, we proceed to construct a solution θ ǫ of (19) which has the desired frequency supportproperty and belongs to the smooth category. Then, by the uniqueness of strong solutions, we pass to thelimit ǫ → (cid:26) ∂ t θ ǫ ( x , t ) − ǫ ∆ θ ǫ ( x , t ) = − M [ θ ǫ ]( x , t ) θ ǫ ( x ,
0) = θ ( x ) (20)and (cid:26) ∂ t θ ǫn +1 ( x , t ) + v ( x , t ) · ∇ θ ǫn ( x , t ) − ǫ ∆ θ ǫn +1 ( x , t ) = − M [ θ ǫn +1 ]( x , t ) θ ǫn +1 ( x ,
0) = θ ( x ) (21)for all n ≥
1. We note that the solutions of (20) and (21) respectively, may be written explicitly using theDuhamel’s formula: θ ǫ ( x , t ) = e − ( ǫ ( − ∆)+ M ) t θ ( x ) ,θ ǫn +1 ( x , t ) = e − ( ǫ ( − ∆)+ M ) t θ ( x ) + ˆ t e − ( ǫ ( − ∆)+ M ) ( t − τ ) ( v ( x , τ ) · ∇ θ ǫn ( x , τ )) dτ. Since e − ( ǫ ( − ∆)+ M ) t is given explicitly by the Fourier multiplier with non-zero symbol e − ( ǫ | k | + d M ( k )) t , thisoperator does not alter the frequency support of the function on which it acts. Therefore, it follows directlyfrom our assumption on the frequency support of θ that supp( b θ ǫ ( t )) ⊂ X C for all t ∈ [0 , T ) . Now, we proceed inductively and note that if supp( c θ ǫn ( t )) ⊂ X C for all t ∈ [0 , T ). Then, by our assumptionon the frequency support of v and Lemma 2.5 we also have supp( \ v · ∇ θ ǫn ( t )) ⊂ X C for all t ∈ [0 , T ). Hence,we obtain that supp( [ θ ǫn +1 ( t )) ⊂ X C for all t ∈ [0 , T ) concluding the proof of the induction step. This provesthat the frequency support of all the iterates θ ǫn ( t ) lies on X C for all t ∈ [0 , T ).Thus, it is left to prove that the sequence { θ ǫn } n ≥ converges to a function θ ǫ which lies in the smoothnessclass (18). Note that there is no cancellation of the highest order term in the nonlinearity. However, since(at least for now) ǫ ∈ (0 ,
1] is fixed, we may use the full smoothing power of the Laplacian.To prove it, for all n ≥ R n ( t ) := sup τ ∈ [0 ,t ] || Λ s θ ǫn || L ( τ ) + ˆ t || p M [Λ s θ ǫn ] || L ( τ ) dτ + ǫ ˆ t || Λ s +1 θ ǫn || L ( τ ) dτ. Moreover, as the frequency support of all the iterates θ ǫn lies on X C ⊂ K C , using Corollary 2.4 we have: || p M [Λ s θ ǫn ] || L ( τ ) ≈ || Λ s θ ǫn || L ( τ ) . In the first step, note that from (20) it follows that for any t ∈ (0 , T ] we obtain that R ( t ) ≤ || Λ s θ || L .We proceed inductively and assume that there exists a time T ⋆ ∈ (0 , T ] such that R n ( T ⋆ ) ≤ || Λ s θ || L .Here, we show that if T ⋆ is chosen appropriately, in terms of θ , v and ǫ , we have R n +1 ( T ⋆ ) ≤ || Λ s θ || L too.From (21), the divergence-free velocity field ∇ · v = 0, integration by parts and the fact that s > / H s an algebra, we obtain: ∂ t || Λ s θ ǫn +1 || L ( t ) + || p M [Λ s θ ǫn +1 ] || L ( t ) + ǫ || Λ s +1 θ ǫn +1 || L ( t ) ≤ || Λ s ( v θ ǫn ) || L ( t )2 ǫ + ǫ || Λ s +1 θ ǫn || L ( t )2and in consequence, as R n ( T ⋆ ) ≤ || Λ s θ || L we have proved that: R n +1 ( T ⋆ ) ≤ || Λ s θ || L + C s ǫ ˆ T ⋆ || Λ s v || L ( τ ) || Λ s θ ǫn || L ( τ ) dτ ≤ || Λ s θ || L + 2 C s T ⋆ ǫ || v || L ∞ (0 ,T ; H s ) || Λ s θ || L . Hence, if we let T ⋆ ≤ ǫ C s || v || L ∞ (0 ,T ; H s ) (22) we have that R n +1 ( T ⋆ ) ≤ || Λ s θ || L . Since T ⋆ is independent of n , it is clear that the inductive argumentmay be carried through, and hence R n ( T ⋆ ) ≤ || Λ s θ || L for all n ≥ n . Taking the difference of two iterates: (cid:26) ∂ t ( θ ǫn +1 − θ ǫn )( x , t ) + v ( x , t ) · ∇ ( θ ǫn − θ ǫn − )( x , t ) − ǫ ∆( θ ǫn +1 − θ ǫn )( x , t ) = − M [ θ ǫn +1 − θ ǫn ]( x , t )( θ ǫn +1 − θ ǫn )( x ,
0) = 0 (23)for all n ≥
2. Similarly to the above, it follows from (23) that e R n ( t ) := sup τ ∈ [0 ,t ] || Λ s ( θ ǫn +1 − θ ǫn ) || L ( τ ) + ˆ t || p M [Λ s ( θ ǫn +1 − θ ǫn )] || L ( τ ) dτ + ǫ ˆ t || Λ s +1 ( θ ǫn +1 − θ ǫn ) || L ( τ ) dτ ≤ C s ǫ ˆ t || Λ s v || L ( τ ) || Λ s ( θ ǫn − θ ǫn − ) || L ( τ ) dτ ≤ t C s ǫ || v || L ∞ (0 ,T ; H s ) e R n − ( t ) for all n ≥ . In particular, due to our choice of T ⋆ ∈ (0 , T ] on (22) we have that e R n ( T ⋆ ) ≤ e R n − ( T ⋆ ), which impliesthat the sequence { θ ǫn } n ≥ is not only bounded, we actually have a contraction in L ∞ (0 , T ⋆ ; H s ( T )) ∩ ǫL (0 , T ⋆ ; H s +1 ( T )) . (24)Hence there exists a limiting function θ ǫn → θ ǫ in the category (24). In addition, since for every n ≥ c θ ǫn ( t )) ⊂ X C and the set X C is closed, we automatically obtain that supp( b θ ǫ ( t )) ⊂ X C .To show that θ ǫ may be continued in (24) up to time T , we note that || Λ s θ ǫ || L ( T ⋆ ) ≤ || Λ s θ || L thanksto the fact that R n ( T ⋆ ) ≤ || Λ s θ || L for all n ≥
1. Hence, repeating the above argument with initialcondition θ ǫ ( T ⋆ ), we obtain a solution θ ǫ ∈ L ∞ (0 , T ⋆ ; H s ( T )) ∩ ǫL (0 , T ⋆ ; H s +1 ( T )) whit the bound || Λ s θ ǫ || L (2 T ⋆ ) ≤ || Λ s θ ǫ || L ( T ⋆ ) ≤ || Λ s θ || L .The above argument may be extended iteratively, thereby concluding the construction of the solution θ ǫ in the category (18).In order to close the proof we need to pass to the limit as ǫ →
0. By construction we have that θ ǫ isuniformly bounded, with respect to ǫ in L ∞ (0 , T ; H s ( T )), and from (19) we obtain that ∂ t θ ǫ is uniformlybounded, with respect to ǫ in L ∞ (0 , T ; H s − ( T )) ∩ L (0 , T ; H s − ( T )). In particular, from the uniformbounds (with respect to ǫ ) of θ ǫ and ∂ t θ ǫ in the corresponding norms, one can use the Banach-Alaoglutheorem and the Aubin-Lions’s compactness lemma (see, e.g. [20] or [28]) to justify that one can extract asubsequence of θ ǫ and ∂ t θ ǫ (using the same index for simplicity) as ǫ → θ and ∂ t θ , such that: • θ ǫ → θ strongly in C (0 , T ; H s − ( T )). • ∂ t θ ǫ ⇀ ∂ t θ weakly in L (0 , T ; H s − ( T )). • ∂ t θ ǫ ∗ ⇀ ∂ t θ weakly- ∗ in L ∞ (0 , T ; H s − ( T )).Now, from (19) we have that ∂ t θ ǫ → − M [ θ ] − v · ∇ θ in C (0 , T ; H s − ( T )). Moreover, as θ ǫ → θ in C (0 , T ; H s − ( T )), the distribution limit of ∂ t θ ǫ must be ∂ t θ for the closed graph theorem [1]. In consequence,since the evolution is linear and s is large enough, it follows that this limiting function is the unique smoothsolution of (17) which lies in L ∞ (0 , T ; H s ( T )). Lastly, since for every ǫ ∈ (0 ,
1] we have supp( b θ ǫ ) ⊂ X C ,and since X C is closed, we obtain that the limiting function also has the desired support property, i.e.supp( b θ ) ⊂ X C , which concludes the proof of the theorem. (cid:3) The main difficulty in the proof of the previous theorem is the construction of an iteration scheme which isboth suitble for energy estimates and preserves the feature that in each iteration step the frequency supportof the approximation lies on X C . Now, we are ready to prove the main result of this section. Proof of Theorem 3.1.
In order to construct the local in time solution θ with frequency support in X C , weconsider the sequence of approximations { θ n } n ≥ given by the solutions of (cid:26) ∂ t θ ( x , t ) = − M [ θ ]( x , t ) θ ( x ,
0) = θ ( x ) (25)and ∂ t θ n ( x , t ) + u n − ( x , t ) · ∇ θ n ( x , t ) = − M [ θ n ]( x , t ) u n − ( x , t ) = M [ θ n − ]( x , t ) θ n ( x ,
0) = θ ( x ) (26) N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 13 for all n ≥
2. One may solve (25) explicitly in the frequency space as b θ ( k , t ) = e − d M ( k ) t b θ ( k ) for k ∈ Z .Hence, it is clear that supp( b θ ( t )) ⊂ supp( b θ ) ⊂ X C for all t ≥
0. Moreover, we have that: || Λ s θ || L ( t ) + 2 ˆ t || p M [Λ s θ ] || L ( τ ) dτ = || Λ s θ || L for all t ≥ . In particular, fixed
T >
0, we obtain the bound: || Λ s θ || L ∞ (0 ,T ; L ) + || p M [Λ s θ ] || L (0 ,T ; L ) ≤ || Λ s θ || L . In order to solve (26) we appeal to Theorem 3.2. Indeed, by the inductive assumption we have that θ n − ∈ L ∞ (0 , T ; H s ) and also that supp( [ θ n − ( t )) ⊂ X C for all t ∈ [0 , T ). Hence, as u n − ≡ M [ θ n − ] by applyingLemma 2.2 we have that u n − ∈ L ∞ (0 , T ; H s ) and by Lemma 2.5 we have supp( [ u n − ( t )) ⊂ X C for t ∈ [0 , T ).Therefore, all the conditions of Theorem 3.2 are satisfied, by letting v = u n − , and there exists a uniquesolution θ n ∈ L ∞ (0 , T ; H s ) of (26), such that supp( c θ n ( t )) ⊂ X C for t ∈ [0 , T ). Moreover, using that θ haszero vertical mean on T the sequence { θ n } n ≥ satisfies the same by construction.To prove that the sequence { θ n } n ≥ converges, we first prove that it is bounded. To do it, we assumeinductively that the following bound holds for all 1 ≤ j ≤ n − j = n . || Λ s θ j || L ∞ (0 ,T ; L ) + || p M [Λ s θ j ] || L (0 ,T ; L ) ≤ || Λ s θ || L . (27)Applying Λ s to (26) and taking an L inner product with Λ s θ n we obtain: ∂ t || Λ s θ n || L ( t ) + || p M [Λ s θ n ] || L ( t ) ≤ || Λ s θ n || L ( t ) || [ u n − · ∇ , Λ s ] θ n || L ( t ) . || Λ s θ n || L ( t ) ( ||∇ u n − || L ∞ || Λ s θ n || L + || Λ s u n − || L ||∇ θ n || L ∞ ) ( t )and for s > / ∂ t || Λ s θ n || L ( t ) + || p M [Λ s θ n ] || L ( t ) . || Λ s θ n || L ( t ) || Λ s u n − || L ( t ) . (28)Above, we have used the fact that ∇ · u n − = 0 in order to write the commutator estimate and the Sobolevembedding L ∞ ( T ) ֒ → H / + ( T ). Since u n − is obtained from θ n − by a bounded Fourier multiplier (cf.Lemma 2.2) there exists a positive constant m ⋆ such that: || Λ s u n − || L ( t ) ≤ m ⋆ || Λ s θ n − || L ( t ) . (29)In consequence, putting together (28) and (29) for s > / ∂ t || Λ s θ n || L ( t ) + || p M [Λ s θ n ] || L ( t ) ≤ C s m ⋆ || Λ s θ n || L ( t ) || Λ s θ n − || L ( t ) . By Corollary (2.4) there exists two positive constants such that m ⋆ ≤ c M ( k ) ≤ m ⋆ for all k ∈ Z ⋆ . Hence,applying H¨older’s inequality we get: ∂ t || Λ s θ n || L ( t ) + || p M [Λ s θ n ] || L ( t ) ≤ C s m ⋆ √ m ⋆ || Λ s θ n || L ( t ) || Λ s θ n − || L ( t ) || p M [Λ s θ n ] || L ( t ) ≤ (cid:18) C s m ⋆ √ m ⋆ (cid:19) || Λ s θ n || L ( t ) || Λ s θ n − || L ( t ) + 12 || p M [Λ s θ n ] || L ( t ) . Using the inductive assumption (27), it follows for t ∈ [0 , T ) that: ∂ t || Λ s θ n || L ( t ) + || p M [Λ s θ n ] || L ( t ) ≤ (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) || Λ s θ n || L ( t )and applying Gr¨onwall’s inequality, we arrive to: || Λ s θ n || L ( t ) + ˆ t || p M [Λ s θ n ] || L ( τ ) dτ ≤ exp "(cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) t || Λ s θ || L . Therefore, taking T ≤ log 2 (cid:16) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:17) (30)we obtain that (27) holds for j = n and so by induction it holds for all j ≥
1. This shows that the sequence { θ n } n ≥ is uniformly bounded in L ∞ (0 , T ; H s ) . Moreover, we may show that the sequence { θ n } n ≥ is Cauchy in L ∞ (0 , T ; H s − ). To see this, we considerthe difference of two iterates e θ n := θ n − θ n − . It follows from (26) that e θ n is a solution of: ∂ t e θ n ( x , t ) + u n − ( x , t ) · ∇ e θ n ( x , t ) + e u n − ( x , t ) · ∇ θ n − ( x , t ) = − M [ e θ n ]( x , t ) u n − ( x , t ) = M [ θ n − ]( x , t ) e θ n ( x ,
0) = 0 (31)for all n ≥
3, where e u n ( x , t ) := M [ e θ n ]( x , t ). Applying Λ s − to (31), taking an L inner product with Λ s − and using that ∇ · u n − = 0, we arrive to: ∂ t || Λ s − e θ n || L ( t ) + || p M [Λ s − e θ n ] || L ( t ) ≤ || Λ s − e θ n || L ( t ) || (cid:2) u n − · ∇ , Λ s − (cid:3) e θ n || L ( t )+ || Λ s − e θ n || L ( t ) || Λ s − ( e u n − · ∇ θ n − ) || L ( t ) . (32)Now, applying the Sobolev embeddings into the product and commutator estimate given by Lemma 1.1 andLemma 1.2, we obtain for s > / • || (cid:2) u n − · ∇ , Λ s − (cid:3) e θ n || L . ||∇ u n − || L ∞ || Λ s − ∇ e θ n || L + || Λ s − u n − || L ||∇ e θ n || L . || Λ / + u n − || L || Λ s − e θ n || L + || Λ s u n − || L || Λ / + e θ n || L . || Λ s u n − || L || Λ s − e θ n || L . • || Λ s − ( e u n − · ∇ θ n − ) || L . || e u n − || L ∞ || Λ s − ∇ θ n − || L + || Λ s − e u n − || L ||∇ θ n − || L ∞ . || Λ / + e u n − || L || Λ s θ n − || L + || Λ s − e u n − || L || Λ / + θ n − || L . || Λ s − e u n − || L || Λ s θ n − || L . Combining these two inequalities with Lemma 2.2 and Corollary 2.4 we have proved that there exists twopositive constants satisfying m ⋆ ≤ c M ( k ) ≤ m ⋆ for all k ∈ Z ⋆ such that: || (cid:2) u n − · ∇ , Λ s − (cid:3) e θ n || L + || Λ s − ( e u n − · ∇ θ n − ) || L . m ⋆ || Λ s θ n − || L (cid:16) || Λ s − e θ n || L + || Λ s − e θ n − || L (cid:17) . Hence, combining this estimate with (32), we arrive to: ∂ t || Λ s − e θ n || L + || p M [Λ s − e θ n ] || L ≤ C s m ⋆ || Λ s − e θ n || L || Λ s θ n − || L (cid:16) || Λ s − e θ n || L + || Λ s − e θ n − || L (cid:17) ≤ C s m ⋆ √ m ⋆ || p M [Λ s − e θ n ] || L || Λ s θ n − || L (cid:16) || Λ s − e θ n || L + || Λ s − e θ n − || L (cid:17) ≤ || p M [Λ s − e θ n ] || L + 12 (cid:18) C s m ⋆ √ m ⋆ (cid:19) || Λ s θ n − || L (cid:16) || Λ s − e θ n || L + || Λ s − e θ n − || L (cid:17) and, as consequence of (27) we get: ∂ t || Λ s − e θ n || L ( t ) ≤ (cid:18) C s m ⋆ √ m ⋆ (cid:19) || Λ s θ n − || L ( t ) (cid:16) || Λ s − e θ n || L ( t ) + || Λ s − e θ n − || L ( t ) (cid:17) ≤ (cid:18) C s m ⋆ √ m ⋆ (cid:19) || Λ s θ || L (cid:16) || Λ s − e θ n || L ( t ) + || Λ s − e θ n − || L ( t ) (cid:17) . Hence, applying Gr¨onwall’s inequality (see [6, p. 624]) we obtain that: || Λ s − e θ n || L ( t ) ≤ exp " (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) t || Λ s − e θ n || L (0)+ exp " (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) t (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) ˆ t || Λ s − e θ n − || L ( τ ) dτ and as by definition e θ n ( x ,
0) = 0, we have for 0 ≤ t ≤ T that: || Λ s − e θ n || L ( t ) ≤ || Λ s − e θ n − || L ∞ (0 ,T ; L ) exp " (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) T (cid:18) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:19) T. N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 15
Here, after recalling (30), if we let T be such that: T := min { log 2 , / } (cid:16) C s m ⋆ √ m ⋆ || Λ s θ || L (cid:17) we obtain that: || Λ s − e θ n || L ∞ (0 ,T ; L ) ≤ || Λ s − e θ n − || L ∞ (0 ,T ; L ) . Consequently { θ n } n ≥ is Cauchy in L ∞ (0 , T ; H s − ) and hence θ n converges strongly to θ in L ∞ (0 , T ; H s − ).Nothing that s − > /
2, this shows that the strong convergence occurs in a H¨older space, which is sufficientto prove that the limiting function θ ∈ L ∞ (0 , T ; H s ) is a solution of the initial value problem (14).To conclude the proof of the theorem we prove the uniqueness. We note that if θ (1) and θ (2) are twosolutions of (14), then θ ♯ = θ (1) − θ (2) solves ∂ t θ ♯ ( x , t ) + u (1) ( x , t ) · ∇ θ ♯ ( x , t ) + u ♯ ( x , t ) · ∇ θ (2) ( x , t ) = − M [ θ ♯ ]( x , t ) u ♯ ( x , t ) = M [ θ ♯ ]( x , t ) θ ♯ ( x ,
0) = 0 . (33)An L estimate on (33) shows that θ ♯ ( x , t ) = 0 for all t ∈ [0 , T ), since θ (2) ∈ L ∞ (0 , T ; H / + ) and thefrequency support of θ ♯ belongs to X C , due to the fact that supp( d θ ( i ) ( t )) ⊂ X C for t ∈ [0 , T ) and i = 1 , . (cid:3) Global existence in H / + ( T ) for frequency-localized initial data This section is devoted to prove the main result of this paper:
Theorem 4.1.
Fixed C > and the frequency straight line X C . Let θ ∈ H s ( T ) with zero vertical meanand frequency support in X C such that || θ || H κ ≤ ǫ where ǫ is give by (45) and κ := α +
52 + for α ∈ (0 , .Then, the solution of the non-diffusive MG equation (6) with initial datum Θ( x ,
0) = Ω( x ) + θ ( x ) existsglobally in time and satisfies the following exponential decay to the steady state: || Θ − Ω || H s ( t ) ≡ || θ || H s ( t ) . || θ || H s exp( − m ⋆ t ) . In the next sections we give the proof of this result.4.1.
Energy methods for the MG equation.
For s > / θ ∈ H s ( T ) with zero verticalmean and supp( b θ ) ⊂ X C , there exists T > θ ( t ) ∈ H s ( T ) and supp( b θ ( t )) ⊂ X C for all t ∈ [0 , T ).4.1.1. A priori energy estimates.
In what follows, we assume that θ ( t ) ∈ H s ( T ) is a solution of (14) andthe frequency support supp( b θ ( t )) ⊂ X C for any t ≥
0. Then, the following estimate holds: ∂ t || θ || H s ( t ) . − (cid:2) − C || θ || H / ( t ) (cid:3) || θ || H s ( t ) . First of all, we will perform the basic H s -energy estimate for ∂ t θ ( x , t ) + u ( x , t ) · ∇ θ ( x , t ) = − M [ θ ]( x , t ) (34)where u ( x , t ) = M [ θ ]( x , t ) and the initial data θ ( x ) has zero vertical mean and frequency support in X C . L -estimate: We multiply (34) by θ and integrate over T . Then: ∂ t || θ || L = − ˆ T θ M [ θ ] d x − ˆ T θ ( u · ∇ ) θ d x . Therefore, using Plancherel’s theorem and (15), we obtain that: ∂ t || θ || L = − X k ∈ Z ⋆ c M ( k ) | ˆ θ ( k ) | = −|| p M [ θ ] || L . (35)˙ H s -estimate: Applying Λ s to (34) and taking an L inner product with Λ s θ we obtain: ∂ t || θ || H s = − ˆ T Λ s θ M [Λ s θ ] d x − ˆ T Λ s θ Λ s [( u · ∇ ) θ ] d x = I + I . First of all we study I . As before, by Plancherel’s theorem and the square roor of M given by (15) we get: I = − X k ∈ Z ⋆ c M ( k ) | d Λ s θ ( k ) | = −|| p M [Λ s θ ] || L = −|| p M [ θ ] || H s . (36)Secondly, we study I . Below, we use the fact that ∇ · u = 0 in order to obtain a commutator operator: I = − ˆ T Λ s θ Λ s [( u · ∇ ) θ ] d x ± ˆ T Λ s θ ( u · ∇ )Λ s θ d x = − ˆ T Λ s θ [Λ s , u · ∇ ] θ d x . Hence, using the commutator estimate (9) and the Sobolev embedding L ∞ ( T ) ֒ → H / + ( T ) we arrive to: I ≤ || Λ s θ || L || [Λ s , u · ∇ ] θ || L . || Λ s θ || L (cid:0) ||∇ u || L ∞ || Λ s − ∇ θ || L + || Λ s u || L ||∇ θ || L ∞ (cid:1) . || θ || ˙ H s (cid:0) || u || H / || θ || ˙ H s + || u || ˙ H s || θ || H / (cid:1) . Since supp( b θ ( t )) ⊂ X C as lons as the solution exists, applying Lemma 2.2 we have that the Fourier operator c M ( k ) restricted to k ∈ X C behaves like a zero order operator. In particular, for s > / I ≤ C s m ⋆ || θ || H s || θ || H / (37)where m ⋆ ( C ) blows-up as C tends to infinity. Putting together (36) and (37), for s > / ∂ t || θ || H s ≤ C s m ⋆ || θ || H s || θ || H / − || p M [ θ ] || H s . (38)To sum up, we have proved the next energy estimate. Theorem 4.2.
Let θ ( t ) ∈ H s ( T ) be a solution of (14) with zero mean and supp (ˆ θ ( t )) ⊂ X C for any t ≥ .Then, for s > / the following estimate holds: ∂ t || θ || H s ( t ) ≤ − m ⋆ h − (cid:16) C s m ⋆ m ⋆ (cid:17) || θ || H / ( t ) i || θ || H s ( t ) . Proof.
Putting together (35) with (38) we arrive to: ∂ t || θ || H s ( t ) ≤ C s m ⋆ || θ || H s ( t ) || θ || H / ( t ) − || p M [ θ ] || H s ( t ) . Using that supp(ˆ θ ( t )) ⊂ X C for any t ≥ ∂ t || θ || H s ( t ) ≤ C s m ⋆ || θ || H s ( t ) || θ || H / ( t ) − m ⋆ || θ || H s ( t )where m ⋆ ( C ) goes to zero as C tends to infinity. Rewriting it, we have achieved our goal. (cid:3) So, as consequence, we establish a “small” data global existence result.
Corollary 4.3.
Fixed s > / . Let θ ∈ H s ( T ) with zero vertical mean and supp ( b θ ) ⊂ X C such that || θ || H / ≤ ε small enough. Then, the solution exists globally in time and satisfies a maximum principle: || θ || H s ( t ) ≤ || θ || H s . In the following section we will improve the previous result. Using a perturbative argument, we are ableto derive explicit expressions that quantify the decay rates. This leads to an asymptotic stability result ofthe steady state.4.2.
Linear & non-linear estimates.
The linearized equation gives very good decay properties. Hence,the main achievement of this section is to control the nonlinearity, so that it would not destroy the decayprovided by the linearized equation.4.2.1.
Linear decay.
We approach the question of global well-posedness for a small initial data from a per-turbative point of view, i.e., we see (14) as a non-linear perturbation of the linear problem. The linearizedequation around the trivial solution ( θ, u ) = (0 ,
0) reads as (cid:26) ∂ t θ ( x , t ) + M [ θ ]( x , t ) = 0 θ ( x ,
0) = θ ( x ) (39)where the initial data θ ∈ H s ( T ) has zero vertical mean and frequency support in X C .As M is a positive operator, we derive the exponential decay in time of solutions to the linear problemwith decay rate depending of the frequency support of the initial data. N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 17
Corollary 4.4.
The solution of (39) with initial data θ ∈ H s ( T ) and with zero vertical mean and frequencysupport in X C satisfies that || θ || H s ( t ) ≤ || θ || H s exp( − m ⋆ t ) , where m ⋆ ( C ) goes to zero as C tends to infinity. Non-linear decay.
Next, we will show how this decay of the linear solutions can be used to establishthe stability of the stationary solution ( θ, u ) = (0 ,
0) for the general problem (14). When perturbing aroundit, we get the following system: (cid:26) ∂ t θ ( x , t ) + M [ θ ]( x , t ) = − u ( x , t ) · ∇ θ ( x , t ) θ ( x ,
0) = θ ( x ) (40)where u ( x , t ) = M [ θ ]( x , t ) and the initial data θ satisfies the same hypothesis. Using Duhamel’s formula,we write the solution of (40) as: θ ( x , t ) = e L ( t ) θ ( x , − ˆ t e L ( t − τ ) [ u · ∇ θ ] ( x , τ ) dτ where L ( t ) denotes the solution operator of the associated linear problem (39). Therefore, we have that: || θ || H s ( t ) ≤ || θ || H s exp( − m ⋆ t ) + ˆ t || u · ∇ θ || H s ( τ ) exp ( − m ⋆ ( t − τ )) dτ. (41)4.3. The bootstraping.
We now demonstrate the bootstrap argument used to prove our goal. The generalapproach here is a typical continuity argument that has been used successfully in a plethora of other cases.Theorem 4.2 tell us that the following estimate holds for s > / ∂ t || θ || H s ( t ) ≤ − m ⋆ h − (cid:16) C s m ⋆ m ⋆ (cid:17) || θ || H / ( t ) i || θ || H s ( t ) . (42)In the following, let α ∈ (0 ,
1) be a free parameter and κ := α +
52 + . We want to prove that || θ || H / decaysin time. This will allow us to close the energy estimate and finish the proof. We will prove it through abootstrap argument, where the main ingredient is the estimate (42).4.3.1. Exponential decay of || θ || H / . In order to control || θ || H / ( t ) in time, we have the following result. Lemma 4.5.
Assume that || θ || H κ ≤ ǫ and || θ || H κ ( t ) ≤ ǫ for all t ∈ [0 , T ] where κ = α +
52 + with < α < .Then, we have: || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t ) for all t ∈ [0 , T ] . Proof.
Duhamel’s formula (41) give us: || θ || H / ( t ) ≤ || θ || H / exp( − m ⋆ t ) + ˆ t || u · ∇ θ || H / ( τ ) exp ( − m ⋆ ( t − τ )) dτ and using the algebraic properties of Sobolev spaces we have that: || u · ∇ θ || H / . || u || H / || θ || H / . m ⋆ || θ || H / || θ || H / . The last inequality is due to Lemma 2.2 and the fact that supp( b θ ( t )) ⊂ X C as long as the solution exists.Moreover, due to the well-known Gagliardo-Nirenberg interpolation inequality: || θ || H / . || θ || − αH / || θ || αH /α +5 / with 0 < α < || θ || H / ( t ) ≤ || θ || H / exp( − m ⋆ t ) + ˆ t C α m ⋆ || θ || − αH / ( τ ) || θ || αH /α +5 / ( τ ) exp ( − m ⋆ ( t − τ )) dτ. By hypothesis, we have that || θ || H /α +5 / ( t ) ≤ ǫ on the interval [0 , T ]. Then, we obtain that: || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t ) + ˆ t (4 ǫ ) α C α m ⋆ || θ || − αH / ( τ ) exp ( − m ⋆ ( t − τ )) dτ. (43)In particular, there exist 0 < T ⋆ ( α ) ≤ T such that for t ∈ [0 , T ⋆ ] we have that: || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t ) (44) If we restrict to 0 ≤ t ≤ T ⋆ and we apply (44) into (43), we have: || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t ) + (4 ǫ ) C α m ⋆ exp( − m ⋆ t ) ˆ t exp( − (1 − α ) m ⋆ τ ) dτ ≤ ǫ exp( − m ⋆ t ) (cid:20) ǫ C α m ⋆ (1 − α ) m ⋆ (cid:21) . Taking 0 < ǫ < (1 − α ) m ⋆ C α m ⋆ we have proved that: || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t )for all t ∈ [0 , T ⋆ ] and, by continuity, for all t ∈ [0 , T ]. (cid:3) A new boostraping argument.
In order to control || θ || H κ ( t ) in time, we have the following result. Lemma 4.6.
Assume that || θ || H κ ≤ ǫ and || θ || H κ ( t ) ≤ ǫ for all t ∈ [0 , T ] where κ = α +
52 + with < α < .Then, we have that: || θ || H κ ( t ) ≤ ǫ for all t ∈ [0 , T ] . Proof.
Applying Gr¨onwall’s inequality into (42) and Lemma 4.5, for t ∈ [0 , T ] we have that: || θ || H κ ( t ) ≤ || θ || H κ exp (cid:20) − m ⋆ ˆ t (cid:16) − (cid:16) C κ m ⋆ m ⋆ (cid:17) || θ || H / ( τ ) (cid:17) dτ (cid:21) ≤ || θ || H κ exp (cid:18) ǫ C κ m ⋆ m ⋆ (cid:19) . Taking 0 < ǫ < log √ C κ m ⋆ m ⋆ we have proved that || θ || H κ ( t ) ≤ ǫ for all t ∈ [0 , T ] . (cid:3) Therefore, it is natural to define a “smallness” parameter ǫ given by: ǫ := min ( (1 − α )4 C α , log √ C κ ) m ⋆ m ⋆ . (45)In consequence, a straightforward combination of Lemma 4.5 and Lemma 4.6 give us: Corollary 4.7.
Let θ ∈ H κ ( T ) such that || θ || H κ ≤ ǫ with < ǫ ≤ ǫ . Then, for all t ≥ we have that: || θ || H κ ( t ) ≤ ǫ and || θ || H / ( t ) ≤ ǫ exp( − m ⋆ t ) . Exponential decay of || θ || H s with s > . We have proved the exponential decay in time of || θ || H / ( t ).Then, we are in the position to show how the bootstrap can be closed. This is merely a matter of collectingthe conditions established above and showing that they can indeed be satisfied. Lemma 4.8.
Let θ ∈ H s ( T ) with s ≥ κ such that || θ || H κ ≤ ǫ where < ǫ ≤ ǫ . Then, for all t ≥ wehave that: || θ || H s ( t ) . || θ || H s exp( − m ⋆ t ) . Proof.
Applying Gr¨onwall’s inequality into (42) we have: || θ || H s ( t ) ≤ || θ || H s exp (cid:20) − m ⋆ ˆ t (cid:16) − (cid:16) C s m ⋆ m ⋆ (cid:17) || θ || H / ( τ ) (cid:17) dτ (cid:21) . The exponential decay of || θ || H / proved in Corollary 4.7 give us: || θ || H s ( t ) ≤ || θ || H s exp( − m ⋆ t ) exp (cid:18) ǫ C s m ⋆ m ⋆ (cid:19) and as 0 < ǫ ≤ ǫ there exists a constant C = C ( α, s, κ ) such that || θ || H s ( t ) ≤ C || θ || H s exp( − m ⋆ t ) . (cid:3) Funding:
The author is partially supported by Spanish National Research Project MTM2017-89976-P andICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-556.
Acknowledgements:
The author thanks ´Angel Castro and Diego C´ordoba for their valuable comments.The author acknowledges helpful conversations with Susan Friedlander and Roman Shvydkoy.
N THE NON-DIFFUSIVE MAGNETO-GEOSTROPHIC EQUATION 19
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Instituto de Ciencias Matem´aticas (CSIC), C. Nicol´as Cabrera, 13-15, 28049 Madrid, Spain
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