Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3}$$\end{document}

We consider a class of steady solutions of the semi-geostrophic equations on R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document} and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L2(R3,R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}}^3,{\mathbb {R}}^3)$$\end{document} introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3}$$\end{document}. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.


Introduction
The aim of this work is to derive and analyse the linearised semi-geostrophic equation (LSG in what follows) associated to a class of steady solutions of the semi-geostrophic equations (referred to as SG) on the whole space ℝ 3 . We present a well-posedness theory for this equation, and discuss the stability of these steady states. For a given conservative steady solution ∇ of SG corresponding to a time-independent strictly-convex quadratic geopotential , the associated LSG in the conservative perturbation ∇ is the passive transport equation where the vector field is a -dependent Eulerian velocity, is a -dependent matrix, ℱ −1 •ℳ •ℱ is a pseudodifferential operator of order 0 with -dependent symbol , and ℱ is the Fourier transform, defined in (1.18). Although the existence theory in 2 (ℝ 3 , ℝ 3 ) of LSG follows from the theory of strongly continuous semigroups on Banach spaces, we provide a representation formula for the solution, which allows us to work with the solutions explicitly. We present and extend this result to the space of tempered distributions  ′ (ℝ 3 , ℝ 3 ), to allow for the treatment of global plane-wave solutions, which are not integrable functions on the whole space ℝ 3 . We also introduce concepts of stability for this family of steady solutions of the semi-geostrophic equations, by studying the long-term behaviour of wave-like solutions, in analogy with the results of [5] on the Navier-Stokes equations. Finally, we briefly mention how the same methods can be applied to the quasi-geostrophic equations, and discuss similarities and differences between the two theories.

Contributions of this Paper
The main novelty of this paper is the introduction of an analytical framework in which one can analyse the linearised semi-geostrophic equations for steady solutions that are globally defined in space. We consider conservative steady solutions corresponding to quadratic strictly-convex potentials: this choice simplifies the derivation, as we will see in Sect. 2, and it allows us to write the linearised equation in a very explicit manner, namely (1.1). Our choice

The Semi-Geostrophic Equations in Eulerian Coordinates
The semi-geostrophic equations can be written in various coordinate systems, depending on the analytical tools that one wishes to employ (e.g. optimal transport theory). The first formulation of SG in Eulerian coordinates was introduced in the 1950s in [9] and was studied by Hoskins in the 1970s (see [10]). However, the model attracted the interest of the mathematical community only in the late 1990s when Benamou and Brenier introduced a new formulation in [4], in a set of coordinates which are now known as geostrophic or dual coordinates, and showed that tools from the theory of optimal transport could be applied to study the well-posedness of the problem. Since then, the problem is typically studied in geostrophic coordinates, because of its connection to optimal transport theory: see, among the others, [1] and [2]. We invite the reader to see [11] for a review of the main formulations and results in the semi-geostrophic theory. For our purposes, we shall consider SG in Eulerian coordinates, which is the original setting in which the equations were derived and studied by Hoskins [10]  in the unknown time-dependent conservative vector field ∇ ∶ Ω × ℝ → ℝ 3 , where Ω ⊆ ℝ 3 . In the above, the matrix ∈ ℝ 3×3 is given by and the function id Ω ∶ ↦ denotes the identity function on Ω. The operator ∶ ∇ ↦ is the formal solution operator associated to the boundary-value problem for the div-curl system on Ω given by This boundary value problem is typically endowed with the usual no-slip condition ⋅ = 0 on Ω if Ω is a bounded domain in ℝ 3 . To guarantee well-posedness of in the case when Ω = ℝ 3 , further conditions must be imposed on . In this article, we are interested in the linear dynamics of SG around a special family of steady solutions in the whole space case Ω = ℝ 3 . This family of steady solutions is realised by the gradients of globally-supported quadratic strictly-convex functions: for a given symmetric positive definite matrix ∈ Sym + (3, ℝ) ⊂ ℝ 3×3 , we consider those steady solutions ∇ of (1.2) with Indeed, one can show that ∇ is a steady solution of (1.2) with corresponding velocity field ∶= [∇ ] given by where the matrix is given by ∶= −1 ( − ) and ∈ ℝ 3×3 is the identity matrix.

Cullen's Stability Principle
The choice of steady solutions (1.5) allows us to derive the linearised SG by formally taking Fréchet derivatives of (1.2) and (1.4) at ∇ in the direction of a smooth conservative vector field ∇ (see Sect. 2). Using the fact that the strictly-convex has constant Hessian, we will show that LSG can be written as the abstract Cauchy problem where the linear operator ℒ is defined as as in (1.1). In the above, the operator ℳ is the multiplication operator ℳ [ ] = with symbol ∶ ℝ 3 ⧵{0} → ℝ which depends on the choice of the matrix and is given by The primary reason we work with this specific family of steady solutions is due to the well-known Stability Principle of Cullen, which is adopted by most authors when analysing SG. Indeed, Cullen's stability principle (introduced for the first time in [7]) postulates that physically stable solutions of SG are those for which (⋅, ) is a convex function in its spatial variable for any . For more about the stability principle, we invite the reader to see [8,Ch. 3] and [6].
As an interesting consequence of our analysis in this paper, we show that whilst steady solutions of the shape (1.5) are stable in the sense of Cullen, they are in fact dynamically unstable in natural topologies. An additional structural benefit to considering steady states of the form (1.5), we show in Sect. 2, is that they result in a linearised equation that involves a pseudo-differential operator of order 0 on ℝ 3 . We do not investigate the linearisation of SG around other steady solutions in this paper, and we leave it for future work. Let us also mention that the corresponding problem in bounded domains Ω ⊂ ℝ 3 is substantially harder, with the existence of non-trivial steady solutions of SG (1.2) on bounded domains being an open problem.

Characterisation of Steady Quadratic Flows
We observed that the velocity field induced by a geopotential of the form (1.5) is given by the linear transformation in (1.6). In particular, different choices for the positive-definite symmetric matrix will produce different steady Eulerian flow fields. To show this, we consider a generic symmetric matrix (1.10) with the coefficients , , , , , ∈ ℝ chosen in such a way that is positive definite. Then, the steady flow velocity is given by ( ) = , with = −1 ( − ). In particular, can be written in terms of the coefficients of and it is easily verified that its spectrum is Hence, the matrix has real eigenvalues if and only if ≥ 0, and it has two purely imaginary eigenvalues otherwise. We define the two sets in the space of positive definite symmetric real matrices As a result, the set + corresponds to hyperbolic flows, whereas the set − corresponds to elliptic flows. Although it is readily seen that the set 0 ∶= { ∈ Sym + (3, ℝ) ∶ = 0} is non-empty, as it contains the identity matrix , we do not investigate the stability of steady solutions corresponding to such matrices.

Statement of Main Results
For reasons pertaining to the study of plane wave perturbations that we outline in the sequel, we aim to construct solutions of (1.1) in  ′ (ℝ 3 , ℝ 3 ) by duality, thereby considering the adjoint LSG given by the abstract Cauchy problem { = , where the operator is the adjoint of the operator ℒ defined in (1.8), namely (1.14) The reader will observe that symbol defined in (1.9) is an even function, therefore a change of variables shows that ℱ•ℳ •ℱ −1 = ℱ −1 •ℳ •ℱ. This means that the analysis of the operator in (1.14) is essentially the same as that of the operator ℒ defined in (1.8), up to a change of sign in the first term and transposing the matrix in the second. In particular, all of the results that we prove for ℒ hold for mutatis mutandis.
Remark 1.1. Since the velocity corresponding to the steady solution (1.5) is given by ( ) = , the associated flow map ( , ) = is a volume-preserving diffeomorphism of ℝ 3 for any time , and we expect LSG in Eulerian coordinates (1.1) to be equivalent to the formulation in Lagrangian coordinates. We will not comment further on the Lagrangian formulation, except for mentioning that the existence of strong solutions of LSG in Lagrangian coordinates can be proved in 2 (ℝ 3 , ℝ 3 ) by applying the theory of evolution systems generated by time-dependent operators, as presented in [12,Ch. 5]. In fact, LSG in Lagrangian coordinates (up to a rescaling) is given by the abstract Cauchy problem where the time-depending operator ( ) is defined as with the symbol̃defined in (1.19) in the next section.
The reason we extend the existence theory of LSG to allow for less regular solutions than those which take values in 2 (ℝ 3 , ℝ 3 ) is so that our solution theory admits plane-wave solutions of the form distribution ( ) = (⋅, ) and characterise the functions and for which is a strong solution of LSG in  ′ , as in the sense that we clarify in Definition 1.7 (v). Then, for these specific solutions, we look at their stability in the following sense.

Notation and Main Definitions
For the reader's convenience, we fix here the notation that will be used throughout.
the weak topology generated by the family of seminorms {‖ ⋅ ‖ , } , ∈ℕ 3 is a Fréchet space. One can prove that the topology generated by the family of seminorms is equivalent to the topology generated by the family of their finite linear combinations. The space  ′ (ℝ 3 , ℝ 3 ) is the space of tempered distributions endowed with the weak*topology. We use the notation  and  ′ in place of (ℝ 3 , ℝ 3 ) and  ′ (ℝ 3 , ℝ 3 ) respectively, unless it is necessary to specify a different range of the Schwartz functions. ℱ and ℱ −1 denote the Fourier transform and its inverse, respectively. To avoid confusion due to the notation, we specify that we define the Fourier transform and its inverse on 1 (ℝ 3 , ℝ 3 ) as follows: and extend this definition to 2 (ℝ 3 , ℝ 3 ) by density in the usual way. We also use the notation̂∶= In order to construct solutions to LSG, we will need to consider the following Fourier symbols, defined in terms of the symbol in (1.9):̃( The operator ℒ defined in (1.8) consists of three additive terms: we will need to refer to them and the groups of operators that they generate, therefore we list here their definitions for the reader's convenience. We formally define the following operators without specifying their domains, as we discuss in Remark 1.6. Definition 1.5. (i) We consider the three additive terms that constitute the operator ℒ = 1 + 2 + 3 defined in (1.8): Remark 1.6. With an abuse of notation, we use the same symbol to denote an operator defined in different topologies, as the topology with which we work will be clear by the context. For instance, we denote by ℒ both the operator in (1.8) defined on the space of distributions  ′ and its restriction to functions in 2 (ℝ 3 , ℝ 3 ) and . By this, we mean that if is a regular distribution corresponding to the function ∈ 2 (ℝ 3 , ℝ 3 ), then we write ℒ = ℒ . For the sake of clarity, we mention that by regular distribution we mean a distribution ∈  ′ such that there exists a function ∈ (ℝ 3 , Some concepts of the theory of semigroups will be employed throughout the paper: for instance, the reader can observe that 1 is the infinitesimal generator of the strongly continuous group The theory of semigroups on Banach spaces is classical (and can be found in [12]), whilst the theory of semigroups on locally convex topological spaces can be found in [15,Ch. IX] and in [3].
We now introduce the notions of solutions that will be of our interest in the following sections.
and it satisfies the differential equation and the initial condition for any ∈  and ∈ ℝ. The time derivative in the expression above is to be interpreted in the weak*-sense in  ′ .
and ∈ 1 (ℝ, ℝ 3 ) such that is of the form (1.16) and the time-dependent regular distribution (⋅, ) is a strong solution in  ′ of LSG corresponding to the regular initial datum (⋅,0) , according to the definition (iv) above.

Structure of the Paper
We present the proofs of the theorems above in the following sections, proceeding as follows: in Sect. 2, we present the formal derivation of LSG for a conservative vector field, starting from SG as written in (1.2), and we comment on the degeneracy of the symbol , distinguishing between the matrices that give rise to a trivial pseudo-differential operator and those that do not. In Sect. 3, we start by presenting the existence theory for solutions in 2 (ℝ 3 , ℝ 3 ), and then provide the existence result in the space of Schwartz functions . We conclude the section with the Proof of Theorem 1.2 and discuss the regularity of solutions with regular initial datum, connecting the statement with the existence in 2 (ℝ 3 , ℝ 3 ). In Sect. 4, we focus on plane-wave solutions of LSG and look at their long-term behaviour in the case of both elliptic and hyperbolic flows, proving Theorem 1.4. In Sect. 5, we briefly investigate the stability of the same family of steady solutions in the quasi-geostrophic theory, and draw a comparison of the results in the two cases.

Derivation of LSG
We provide a formal derivation of LSG in Eulerian coordinates, as in (1.1), from SG in the form presented in (1.2). We observe that SG can be written as the abstract Cauchy problem where the nonlinear operator is defined formally as Proof. Formally, the Fréchet derivative of the operator at ∇ in the direction ∇ is given by We start looking at 1 : by considering the Fourier transform of 1 , namelŷ one can write the explicit formula for 1 : Hence, one can calculate Similarly, to find 2 , we first write2 and compute the inverse Fourier transform to find the explicit formula for 2 given by Therefore, we find We have now a representation formula for ∇ by the expression above and (2.6): To obtain (1.1), we just need to observe that (Degeneracy of the pseudo-differential operator) The flow of LSG is interesting due to the presence of the pseudo-differential operator ℱ −1 •ℳ •ℱ, so we show that the action of this operator is not trivial, i.e. we show that the pseudo-differential operator does not vanish for all choices of the matrix . In fact, we have that We denote the complement of ℬ by ∶= Sym + (3, ℝ)⧵ℬ. An example of a matrix in the set ℬ is = , that describes a non-trivial elliptic flow when ∈ (0, 1) ∪ (1, ∞): indeed, the corresponding steady geopotential and steady flow velocity field are given respectively by In particular, the matrices in the set ℬ can generate both elliptic and hyperbolic flows, defined in (1.12).

Existence Theory for LSG
In this section, we present the existence results for LSG in 2 (ℝ 3 , ℝ 3 ), then in , and finally in  ′ , where we define strong solutions by duality. In fact, as we mentioned before, the operator ℒ defined in (1.8) and the operator ℒ ′ = defined in (1.14) are the same up to a sign and the transposition of the matrix , therefore any result for LSG (1.7) can be mirrored for the abstract Cauchy problem (1.13).
Although the existence of solutions in 2 (ℝ 3 , ℝ 3 ) can be proved by using the theory of strongly continuous semigroups on Banach spaces, we give a direct proof using the explicit representation formula of the solution. Formally, given the initial datum 0 , the solution can be written as (⋅, ) = ( ) 0 , where the operator ( ) is defined in (1.21). We provide here a formal derivation of the formula.
First of all, we derive LSG in Lagrangian coordinates, which we introduced in (1.15). Given a smooth solution of LSG (1.7), we define the function and observe that The derivation above is formal, and we now prove that ( ) 0 is indeed a solution of LSG in the appropriate topologies, starting with 2 (ℝ 3 , ℝ 3 ).

Existence of Solutions of LSG in (ℝ , ℝ )
We now focus on the existence of solutions of LSG in the Fréchet space : as we mentioned in the introduction, we endow  with the weak topology generated by the seminorms (1.17). In order to construct strong solutions of LSG in , one way would be to directly apply the theory of locally equicontinuous semigroups on locally convex topological spaces by showing that 1 , 2 and 3 are infinitesimal generators of locally equicontinuous 0 -groups, and then apply the Trotter formula (see [3,Theorem 20]). Instead, we use the representation formula (1.21) and show that this gives a strong solution in , as by Definition 1.7. In order to do so, we will need estimates for the families of operators { 1 ( )} ∈ℝ , { 2 ( )} ∈ℝ and { 3 ( )} ∈ℝ , in a similar fashion as for the 2 -theory.

( )( ) ,
and ≤ for , ∈ ℕ 3 means that ≤ for ∈ {1, 2, 3}. In particular, the following bound holds for any ∈  and ∈ ℝ, with constants and depending only on , ∈ ℕ 3 . The continuity of the map ↦ 3 ( ) follows from the fact that the symbol̃is smooth in . The estimates (3.6), (3.7) and (3.9) prove that the operator ( ) = 1 ( ) 2 ( ) 3 ( ) maps  into , and that for any , ∈ ℕ 3 there exist constants = ( , , ) and = ( , , ) and a finite set of multi-indices for any 0 ∈ . The continuity of the map ↦ ( ) 0 from ℝ to  follows from the continuity of the operators 1 ( ), 2 ( ) and 3 ( ) in . By Lemma 3.2, we can write ( ) 0 = ( )[ℒ 0 ], and therefore the strong solution is smooth is . A crucial point for the bootstrapping argument is that the operator ℒ maps  into . □ As we mentioned above, the adjoint LSG problem (1.13) is mathematically equivalent to LSG (1.7), therefore the Proof of Theorem 3.3 can be adapted to show the following result.
for any ∈ .

Existence of Solutions of LSG in  ′ (ℝ , ℝ )
The existence of strong solutions of the adjoint LSG (1.13) in  allows us to construct strong solutions to LSG in  ′ .
Proof of Theorem 1.2. For any ∈ ℝ, we consider the operator ( ) defined in Corollary 3.4, and we prove that ↦ ( ) ′ 0 is a strong solution of LSG in  ′ , in the sense of the Definition 1.7. With ( ) ′ we denote the operator on  ′ defined by duality as for a distribution ∈  ′ . In fact, the operator ( ) ′ can be seen as an "extension" of the operator ( ) to the space of tempered distributions. In order to prove that ( ) ′ 0 is a strong solution of LSG, we apply the equivalent of Lemma 3.2 for the operators and ( ): in fact, for any ∈  ′ and ∈ ℝ, the following sequence of identities ⟨ ( ) ′ , holds for any test function ∈ . Continuous differentiability of the function ↦ ( ) ′ 0 from ℝ to  ′ is a consequence of the continuous differentiability of ↦ ( ) 0 from ℝ to  for any 0 ∈ , as proved in Corollary 3.4. □ As we mentioned, ( ) ′ is formally an extension of ( ) to  ′ . This is clarified by the following corollary, that connects the 2 theory of Theorem 3.1 and the  ′ theory of Theorem 1.2. Proof. The definition of the solution that was constructed in the Proof of Theorem 1.2 and a standard calculation allow us to write for any test function ∈  and for any ∈ ℝ By Theorem 3.1, we have that ( ) 0 is a weak (strong) solution in 2 (ℝ 3 , ℝ 3 ) of LSG if 0 ∈ 2 (ℝ 3 , ℝ 3 ) ( 0 ∈ (ℒ)), and if 0 is conservative so is ( ) 0 for all times ∈ ℝ by Theorem 3.1. □

Stability
As we have an existence theory of a global-in-time solution of LSG, we can study the long-time behaviour of its solutions and comment on stability of the steady solutions introduced in (1.5). A comprehensive theory of stability of such solutions is not the purpose of this work, and we instead consider the particular case of plane-wave perturbations. We begin by characterising the specific forms of the amplitude and frequency that give rise to solutions of LSG of the form (1.16), writing their form explicitly in terms of their initial values (0) and (0). We then look at their stability, presenting a Proof of Theorem 1.4.

Plane-wave Solutions
We now consider a specific type of solutions to LSG: we are interested in solutions of the form (1.16). As such functions are not square-integrable on ℝ 3 for any time ∈ ℝ, the concept of plane-wave solutions is clarified in Definition 1.7 (v). It is clear that not all functions of the form (1.16) are plane-wave solutions: indeed, we first characterise the functions and that generate plane-wave solutions.

Stability of the Plane-wave Solutions
We proceed now to prove Theorem 1.4, distinguishing between elliptic and hyperbolic flows, as we defined them in (1.12).
Proof of Theorem 1.4. We use different arguments for the two types of flows, so we look at one case at the time.
We recall the partition Sym + (3, ℝ) = ∪ ℬ introduced in (2.7). I. Hyperbolic flows. For a matrix ∈ + ∩ , the spectrum of is {0, , − }, with > 0, as we observed in (1.11). We denote with ± 0 an eigenvector of corresponding to the eigenvalue ± , and observe that̃( ; ± 0 ) = ±2 : in fact, This allows us to writẽ( ; ± 0 ) = ±2 and to estimate the ∞ -norm of the solution at all times: Hence, if we choose the solution corresponding to an initial datum with 0 eigenvector corresponding to the positive eigenvalue, the solution grows exponentially in time. Likewise, if 0 is chosen in the eigenspace corresponding to the negative eigenvalue, the solution decays exponentially in time.
Since ( ) is -periodic, the integral of ( ( )) can be partitioned as below therefore the integral of ( ( )) over [0, ] is oscillatory, proving that is bounded for any initial data 0 and 0 : We need to prove that (4.2) holds for any initial data 0 and 0 . First of all, we observe that we can focus on the numerator of the symbol defined in (1.9): if 0 ≠ 0, then ( ) ≠ 0 for all , and there exists a constant > 1 such that for all ∈ ℝ 0 < 1 ≤ ( ) ⋅ −1 ( ) ≤ < ∞.

Comparison with the Quasi-geostrophic Approximation
We now present a brief comparison between the stability of the family of steady solutions (1.5) which are solutions of both SG and the quasi-geostrophic equations (QG). The two models are obtained as formal asymptotic limits of the 3-D Euler equations for small Rossby number, and the theory presented in this paper can be mirrored for QG following a similar approach. In fact, rather than writing QG in terms of both full and geostrophic velocities (as presented, for instance, in [14]), one can show that the equations can be written in terms of the full velocity [∇ ] and the gradient ∇ as follows: Therefore, the steady solutions ∇ for QG corresponding to ∇ (1.5) are given by with̃= − and a positive definite symmetric matrix. We mention that is not the physical pressure , which is instead given by ( , ) = ( , ) + 2 2 2 3 , for any = ( 1 , 2 , 3 ) ∈ ℝ 3 and ∈ ℝ.