On the Non-Diffusive Magneto-Geostrophic Equation
Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular that the active scalar. In , the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for perturbations with initial data localized in the frequency space. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the norm of the perturbation.
The geodynamo is the process by which the Earth’s magnetic field is created and sustained by the motion of the fluid core, which is composed of a rapidly rotating, density stratified, electrically conducting fluid. The convective processes in the core that produce the velocity fields required for dynamo action are a combination of thermal and compositional convection. The full dynamo problem requires the examination of the full 3D partial differential equations governing convective, incompressible magnetohydrodyamics (MHD).
It is therefore reasonable to attempt to gain some insight into the geodynamo by considering a reduction of the full MHD equations to a system that is more tractable, but one that retains many of the essential features of the problem. Recently Loper and Moffat ,  proposed the magneto-geostrophic equation (MG) as a model for the geodynamo which is a reduction of the full MHD system.
The nonlinear effects in this three-dimensional system are incorporated in an evolution equation with singular drift for a scalar buoyancy field. An explicit operator encodes the physics of the geodynamo and produces the divergence free drift velocity from the scalar buoyancy field .
1.1. The MG-equation:
We refer to ,  and  for a rigorous justification of the model. To sum up, after a series of approximations relevant to the Earth’s fluid core, a linear relationship is established between the velocity and magnetic vector fields, and the scalar buoyancy .
The sole remaining nonlinearity in the system occurs in the evolution equation for . The active scalar equation for that contains the nonlinear process in Moffatt’s model is precisely
where the divergence-free velocity is explicitly obtained from the bouyancy as where is a non-local differential operator of order 1. We describe below the precise form of that operator.
For simplicity, we consider the domain to be . Without loss of generality we may assume that for all , since the mean of is conserved by the flow. Then, the velocity is obtained from via
for , where the are Fourier multiplier operators with symbols given explicity by
where the Fourier variable . On we let , since for self-consistency of the model we assume that and have zero vertical mean. It can be directly checked that for all (we use the standard summation convention on repeated indices), and hence the velocity field given by (2) is indeed divergence-free.
The physical parameters of the geodynamo are the following: is the rotation rate of the Earth, is the magnetic diffusivity of the fluid core, is the thermal diffusivity, and is the strength of the steady, uniform mean part of the magnetic field in the fluid core. The magnetic field vector is computed from via the operator
In the following, we refer to the evolution equation (1) with singular drift velocity given by (2) as the magneto-geostrophic equation (MG). In addition, we will distinguish between diffusive () and non-diffusive case (). In the Earth’s fluid core the value of the diffusivity is very small. Hence it is relevant to address both the diffusive evolution, and the non-diffusive version where
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 of Hadamard, if given any initial data in a functional space , the problem has a unique solution in , with depending only on the -norm of the initial data, and moreover the solution map satisfies strong continuity properties, e.g. it is uniformly continuous, Lipschitz, or even smooth, for a sufficiently nice space . If one of these properties fail, the Cauchy problem is called ill-posed.
1.1.1. Diffusive vs. non-diffusive MG equation:
Both systems have contrasting properties:
Non-diffusive MG equation: For , in  the authors prove that the equation is ill-posed in the 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 locally well-posed in the class of real-analytic functions in the spirit of a Cauchy-Kowalewskaya result, since each term in the equation loses at most one derivative.
Moreover, in the same article the authors prove that the solution map associated to the Cauchy problem is not Lipschitz continuous with respect to perturbations in the initial data around a specific steady profile for some integer , in the topology of a certain Sobolev space.
The proof consists of a linear and a nonlinear step. After linearizing the problem around , the authors employ techniques from continued fractions in order to construct an unstable eigenvalue for the linearized operator. Once these eigenvalues are exhibited, one may use a fairly robust argument to show that this severe linear ill-posedness implies the Lipschitz ill-posedness for the nonlinear problem.
Hence, without the Laplacian to control the unbounded operator M the situation is dramatically different from the case . For the above, the problem of the fractionally diffusive MG equation arise naturally. This is, one can replace the Laplacian by nonlocal operators, such as for . This situation is non-physical but mathematically interesting, it was addressed in . In the range the equation is locally well-posed, while it is Hadamard Lipschitz ill-posed for . At the critical value the problem is globally well-posed for suitably small initial data, but is ill-posed for sufficiently large initial data.
A further feature of interest is that the anisotropy of the
symbol can be explored to obtain an improvement in the regularity of the solutions when the initial data
is supported on a plane in Fourier space. For such well-prepared data the local existence and uniqueness of solutions can be obtained for all values , and the global existence holds for all initial
data when .
With all this in mind, we seek to determine a steady state around which the non-diffusive MG equation is well-posed. As in , we take advantage of the anisotropy of the symbols (2) and observe an interesting phenomenon: when the initial perturbation is localized 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 smooth as the advected scalar. This enables us to obtain a well-posedness result over the generic setting when no conditions of the Fourier spectrum of the initial perturbation are imposed.
1.1.2. Singular drift
One may view the MG equation as an example of a singular active scalar since the 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 Cauchy problem for the transport equation
where the vector field is related to by some operator. We remark that the MG equation fall into a hierarchy of active scalar equations arising in fluid dynamics in terms of the nature of the operator that produces the drift velocity from the scalar field:
Hierarchy of active scalar equations:
Inviscid MG equation : Singular order 1
: Hadamard ill-posed.
: Critical case, globally well-posed.
Burgers equation (see ): Order 0
: Critical case, globally well-posed.
2D Euler equation (vorticity form) Smoothing degree 1
Viscous MG equation Smoothing degree 2
“Better” than 2D Euler.
The viscous () MG equation, even without thermal diffusion, is âbetter behavedâ than the 2D Euler equation. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity and the scalar temperature produces two orders of smoothing. In  and  the authors study the implications of this property.
1.2. The Fourier multiplier operator
We study the properties and behavior of the Fourier mulplier operator M obtained from (2), which relates and . It is important to note that although the symbols are zero-order homogenous under the isotropic scaling , due to their anisotropy the symbols are not bounded functions of . To see this, note that whereas in the region of Fourier space where the are bounded by a constant, uniformly in , this is not the case on the “curved” frequency regions where and , with . In such regions the symbols are unbounded, since as we have:
In fact, it may be shown that , where is a fixed constant, and this bound is sharp. The fact that symbols are at most linearly unbounded in the whole Fourier space implies that M is the derivative of a singular integral operator of order zero (see  for details). It is clear that any such that may we written as for a skew-symmetric matrix . From now, we just take to be the zero-order fourier multiplier
The matrix corresponding to the MG equation is given by
In particular, the velocity vector is obtained from via
where is a matrix of Calderón-Zygmund singular integral operators (that is, they are bounded in for and ) such that .
Several important properties of the ’s are immediately obvious:
The functions are strongly anisotropic with respect to the dependence on the integers , and . This is a consequence of the interplay of the three physical forces governing this system:
Since the symbols are even the operator M is not anti-symmetric.
These properties of make the MG equation interesting and challenging mathematically, as well as having a clear physical basis in its derivation from the MHD equations.
We emphasize that the mechanism producing ill-posedness is not merely the order one derivative loss in the map . 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 central role in the proof of non-uniqueness for -weak solutions to the non-diffusive MG equation proved in , via methods from convex integration. In contrast, an example of an active scalar equation where the map is unbounded, but given by an odd Fourier multiplier, is the generalized SQG equation where and . This equation was recently shown in  to give a locally well-posed problem in Sobolev spaces.
1.2.1. On the lack of well-posedness for the MG equation in Sobolev spaces
Let us briefly discuss why the evenness of the operator M breaks the classical proof of local existence in Sobolev spaces for the () non-diffusive MG equation. To see why one may not use the standard energy-approach to obtain local well posedness, we point out that in the energy estimate for (1) there are only two terms which seem to prevent closing the estimate at the level:
where we denoted . Since , upon integrating by parts we have . On the other hand, the term does not vanish in general. The only hope to treat the term would be to discover a commutator structure. However, since M is not anti-symmetric, i.e. even in Fourier space, we cannot write , where
If you could do this, a suitable commutator estimate of Coifman-Meyer type would close the estimates at the level of Sobolev spaces. Instead we have that . This is the main reason why we are unable to close estimates at the Sobolev level.
This section contains a few auxiliary results used in the paper. In particular, we recall the, by now classical, product and commutator estimates, as well as the Sobolev embedding inequalities. Proofs of these results can be found for instance in , and .
Lemma 1.1 (Product estimate).
If , then for all we have the estimate
In the case of a commutator we have the following estimate.
Lemma 1.2 (Commutator estimate).
Suppose that . Then for all we have the estimate
where and .
Moreover, the following Sobolev embeddings holds:
continuosly if and .
continuosly if .
It is important to remember that the MG equation is mean preserved. Then, as we said before, we can assume for simplicity that we will always work with periodic functions over with zero mean. Moreover, for the self-consistency of the model (see [11, p. 297]) we restrict to the function spaces where and have zero vertical mean. In fact, without such a restriction the system is not well defined.
It will be useful and it is easy to check that any function with zero vertical mean has immediately zero mean. For this reason, in the rest we work with periodic functions over with zero vertical mean. In particular, note that if has zero mean, then it is clear that: for
1.4. Notation & Organization
We remember that the natural norm in Sobolev spaces is defined by:
For convenience, we may use and to stand for and , respectively. Moreover, to avoid clutter in computations, function arguments (time and space) will be omitted whenever they are obvious from context. Finally, we use the notation when there exists a constant independent of the parameters of interest such that .
Organization of the paper: In Section 2 we begin by setting up the perturbated problem around a specific steady state. We then state a less technical version of the main theorem and we give some general explanation of the ideas behind the proof. Finally, we collect some useful technical lemmas about the behaviour of on suitable subsets of the frequency domain. In Section 3 we embark on the proof of a local existence result for frequency-localized initial data following the ideas of . The core of the article is the proof of the main theorem in Section 4. We commence by the a priori energy estimates given in Section 4.1. This is followed by an explanation of the decay given by the linear semigroup of our system in Section 4.2. Finally, in Section 4.3 we exploit a bootstrapping argument to prove our theorem.
2. The perturbated equation
To sum up, for the non-diffusive MG equation, we have the following system of equations in :
where the operator M is given by (2) and our initial data has zero vertical mean.
2.1. The steady states
When studying fluid equations, it is often helpful to have a good understanding of the exact steady states of the system. The kinds of exact solutions we are interested are the simplest possible steady state, namely and for some -periodic function , with .
The aim of this paper is to show that the Cauchy problem for the non-diffusive MG equation (9) is well-posed with respect to perturbations around a specific steady profile , in the topology of a certain Sobolev space.
In this work, we are interested in the perturbative regime near the special steady state . The main achievement of the paper is a local existence result for perturbations localized in a suitable section of the frequency space together with a global existence result under an additional size condition over the norm of the perturbation.
To sum up, we want to consider solutions in for with the structure:
where has zero vertical mean and frequency support in . Then, we prove:
Local well-posedness: If .
Global well-posedness: If and is small enough.
GWP & asymptotic stability: If and is small enough.
A precise statement of our result is presented as Theorem 4.1, where we also illustrate its proof through a bootstrap argument. Despite the apparent simplicity, understanding the stability of this flow is non-trivial.
2.2. The ideas behind the proof
In order to prove this we first consider the stability of the problem, when linearized about a particular steady state . The main mechanism of decay can be seen from the linearized equation:
As is a positive operator for there is a unique positive self-adjoint square root operator of on , which we define in (11). In consequence, the linearized equaiton clearly shows the decay over time of , except for the zero mode in . However, we do not have that problem because for 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 destroy the decay provided by the linearized equation. Note that, over the curved frequency regions where and with , we have that and with and we can not control and close the estimates at the level of Sobolev spaces. But, as in , we explore the following observation: if the frequency support of lies on a suitable section of the Fourier space, then the operator behaves like an order zero operator and hence the corresponding velocity is as smooth as . This enables us to obtain a well-posedness results over the generic setting when no conditions on the Fourier spectrum of the initial perturbation are imposed. To be more precise, we consider an appropiate subset which we will define later, where we can obtain a local well-posedness result for perturbations such that .
Under this hypothesis over the initial perturbation, at least morally speaking, our perturbated system behaves like an active scalar of order zero with a damping term:
and as , the type of results obtained for the supercritical diffusive MG equation in  are expected to have also in our setting.
2.3. The perturbated MG equation
We denote by the perturbation . Therefore, we obtain the system:
where our initial data has zero vertical mean.
What is interesting about this equation is that is a positive operator so we get a mild dissipation effect. This structure will allow us to prove stability. So, just as for the fractional Laplacian, we define the square root of via Fourier transform as follows:
The square root of can be defined on functions with zero vertical mean as a Fourier multiplier given by the formula:
Note that is not defined on since for the self-consistency of the model, we only work with periodic functions with zero vertical mean.
2.3.1. Well-prepared initial data
In this section we explore the observation cited above: if the frequency support of lies on a suitable subset of the frecuency space, then the operator M is mild when it acts on , i.e. it behaves like an order zero operator, and hence the corresponding velocity is as smooth as .
This enables us to obtain a well-posedness result over the generic setting when no conditions on the Fourier spectrum of the initial perturbation are imposed. For instance, the local existence and uniqueness of smooth solutions holds for the non-diffusive case, a setting in which we know that for generic initial data the problem is ill-posed in Sobolev spaces.
Since we are working in the periodic setting , the frequency space is . Then, let be a positive rational constant. We define in the frequency space the following subsets:
and the union of the above two sets
We shall need the next straightforward observation:
is closed under addition of vectors.
is closed under addition of vectors.
The following lemmas states that M behaves like an zero order operator when it acts on functions with frequency support in . Throughout this paper we shall make key use of the next properties of M.
Let . For every smooth periodic function with zero vertical mean and frequency support in , there exists a universal constant such that:
for all . Moreover, the constant blows-up as tends to infinity.
It is clear that the bound has to be proven only for , since otherwise we have that and the statement holds trivially. We now consider each of the cases .
For , a short algebraic computation gives
from which it follows that for a suitable constant .
Similarly to the previous one, it follows for and that:
This concludes the proof of the lemma. ∎
Let . For every smooth periodic function with zero vertical mean and frequency support in , there exists a universal constant such that:
for all . Moreover, the constant goes to zero as tends to infinity.
It is clear that the bound has to be proven only for , since otherwise we have that and the statement holds trivially. As is a Fourier multiplier operator, we have that:
Morover, for we have the bound
from which it follows that for a suitable constant . ∎
As a consequence of the previous lemmas, under the same hypothesis as before we have that the Fourier operator is equivalent to the identity operator in . More specifically, there exists a pair of real numbers such that:
Let . For every smooth periodic function with zero vertical mean and frequency support in , there exists positive constants and such that:
In the spirit of , fixed we define the frequency straight line to be the set:
Moreover, it follows trivially that:
is closed under addition of vectors.
In consequence, the previous lemmas are also valid for functions with frequency support in for Note that the point does not give any problem because we consider only functions with zero mean.
The key point of the result of well-posedness is the fact that we only work with frequency localized initial perturbations. As we will see later in the proof, to prove that the perturbation does not leave the region of the frequency space where the operator M behaves like a zero order operator, the sets closed under addition will play a crucial role.
2.3.2. Closed sets under addition
A set is closed under addition if for all we have that . In other words, performing the binary operation on any two elements of the set always gives you back something that is also in the set.
In the rest of the paper, fixed we assume that . This is, in the following for we will understand one of the previously defined sets.
Fixed and . For every pair of smooth periodic function with frequency support in we have that:
for all .
The proof is an immediate consequence of the properties of the Fourier transform:
Clearly since is closed under addition.
As M is a Fourier multiplier, we have:
3. Local existence for frequency-localized initial data
The main result of this section is:
Fixed and . Assume that with has zero vertical mean and satisfies that . Then, there exists a time and a unique smooth solution
of the Cauchy problem (10) such that for all .
Before that, the goal is to prove the existence of smooth solutions to the scalar linear equation:
where the initial datum and the given divergence-free drift velocity field satisfies that:
for all for a positive time .
The main result is:
Let . Given and a divergence-free vector field satisfying the above conditions. Then, there exists a unique smooth solution of (12) such that:
Moreover, we have that for all
Proof of Theorem 3.2.
for and finally we pass to the limit in order to obtain a solution of the original system.
On the other hand, we proceed to construct a solution of (14) which has the desired frequency support property and belongs to the smooth category. Then, by the uniqueness of strong solutions, we pass to the limit and obtain a solution with desired properties. We consider the following iterative scheme:
Since is given explicitly by the Fourier multiplier with non-zero symbol , this operator does not alter the frequency support of the function on which it acts. Therefore, it follows directly from our assumption on the frequency support of that for all
Now, we proceed inductively and note that if for all . Then, by our assumption on the frequency support of and Lemma 2.5 we also have for all . Hence, we obtain that for all concluding the proof of the induction step. This proves that the frequency support of all the iterates lies on for all .
Thus, it is left to prove that the sequence converges to a function which lies in the smoothness class (13). Note that there is no cancellation of the highest order term in the nonlinearity. However, since (at least for now) is fixed, we may use the full smoothing power of the Laplacian.
To prove it, for all we define:
Moreover, as the frequency support of all the iterates lies on , using Corollary 2.4 we have:
In the first step, note that from (15) it follows that for any we obtain that . We proceed inductively and assume that there exists a time such that . Here, we show that if is chosen appropriately, in terms of and , we have too.
From (16), the divergence-free velocity field , integration by parts and the fact that which makes an algebra, we obtain: