On the well-posedness of Galbrun’s equation

Galbrun’s equation, which is a second order partial differential equation describing the evolution of a so-called Lagrangian displacement vector field, can be used to study acoustics in background flows as well as perturbations of astrophysical flows. Our starting point for deriving Galbrun’s equation is linearized Euler’s equations, which is a first order system of partial differential equations that describe the evolution of the so-called Eulerian flow perturbations. Given a solution to linearized Euler’s equations, we introduce the Lagrangian displacement as the solution to a linear first order partial differential equation, driven by the Eulerian perturbation of the fluid velocity. Our Lagrangian displacement solves Galbrun’s equation, provided it is regular enough and that the so-called “no resonance” assumption holds. In the case that the background flow is steady and tangential to the domain boundary, we prove existence, uniqueness, and continuous dependence on data of solutions to an initial–boundary value problem for linearized Euler’s equations. For such background flows, we demonstrate that the Lagrangian displacement is well-defined, that the initial datum of the Lagrangian displacement can be chosen in order to fulfill the “no resonance” assumption, and derive a classical energy estimate for (sufficiently regular solutions to) Galbrun’s equation. Due to the presence of zeroth order terms of indefinite signs in the equations, the energy estimate allows solutions that grow exponentially with time.


Introduction
The linearized Euler's equations constitute a standard model for propagation of sound in a background flow. What appears to be less known is that the linearized Euler's equations can be reduced to a vector "wave" equation in the Lagrangian displacement, that is, the displacement of individual fluid particles. (The precise definition of the Lagrangian displacement will be given later.) The resulting equation is often referred to as Galbrun's equation in the literature, in honor of Henri Galbrun who first derived the equation in 1931 [22][ Chapter 3]. Since that first account, Galbrun's equation (or at least very similar equations) has been independently rediscovered and investigated a multiple of times [20,8,23,26,30,19,14], with applications in acoustics and astrophysics.
The linearized Euler's equations are derived from Euler's equations by using an Eulerian linearization ansatz [21]. Analogously, Galbrun's equation may be derived by using a Lagrangian linearization ansatz [21]. However, we will present a complementary derivation, for homentropic background flows, of Galbrun's equation that does not rely on Lagrangian perturbations. One of many possible formulations of Galbrun's equation reads [21,5,17] (note that the last two references assume that ϕ 0 = δϕ = 0) where the vector field w denotes the Lagrangian displacement; u 0 , p 0 , ρ 0 , and c 0 the fluid velocity, pressure, density, and speed of sound fields of the background flow, respectively; ϕ 0 the volume force density acting on the background fluid; δϕ a volume force density; and D 0 = ∂ t + u 0 · ∇ the material derivative with respect to u 0 . Apart from reducing the number of unknowns and equations to be solved, it is pointed out in the literature that Galbrun's equation allows natural handling of boundary conditions, since the primary unknown is the Lagrangian displacement [23,27]. Moreover, Galbrun's equation (1) may be derived using the Euler-Lagrange formalism and thus allows formulation of a wave energy balance law [8,30,23,5].

Previous works on the well-posedness of Galbrun's equation
Naive finite element discretizations of the time harmonic counterpart of Galbrun's equation (1) for steady background flows are known to yield poor numerical results [31,25]. The situation here appears similar to the issues of locking in linear elasticity and approximation of the curl-curl operator in electrodynamics.
In a sequence of papers that consider increasingly complicated background flows, a regularized formulation of Galbrun's equation has been proposed to resolve the numerical issues for subsonic background flows [10,11,9,12,13]. The idea behind the regularization is to take advantage of the identity −∆w = −∇(∇ · w) + ∇ × (∇ × w). For a homogenous background flow, the time harmonic Galbrun's equation at angular frequency ω readŝ whereD 0 = iω + u 0 · ∇, w(x, t) =ŵ(x) exp iωt, and δϕ(x, t) = δφ(x) exp iωt. The regularized formulation of equation (2) is constructed by adding c 2 0 ∇ × (∇ ×ŵ − ψ) to the left hand side of the equation. We note that if ψ = ∇ ×ŵ, the added term vanishes and the time harmonic Galbrun's equation (2) is retrieved. The regularized equation is coupled with an equation for ψ, which is obtained by applying the curl operator to the time harmonic Galbrun's equation (2) and in the end replacing ∇ × w with ψ. Although the commutator term [(iω + u 0 · ∇) 2 , ∇×]ŵ in equation (3) vanishes for a homogeneous background flow, we note that it would be present for more complicated background flows. In order to reconcile the regularized formulation with the original formulation, it is required that ψ = ∇ ×ŵ on the boundary of the domain. It turns out that the regularized formulation of time harmonic Galbrun's equation, with perfectly matched layers to handle artificial boundaries, is well-posed in two spatial dimensions under relatively mild assumptions on the background flow [13]. Nevertheless, a recent numerical study [7] reported lack of convergence of the numerical solution in the case of a heterogeneous background flow; interestingly no such convergence issues were observed when solving linearized Euler's equations. An alternative to the regularized time-harmonic formulation that has been used to generate numerical solutions [31], is based on a mixed variational formulation of the system where δ L p(x, t) = δ Lp (x) exp iωt denotes the Lagrangian pressure perturbation (a precise definition of Lagrangian perturbations are given in the next section). To the best of our knowledge, wellposedness of formulation (4) has not been established. The case of general time dependence appears to have received less attention in the literature than its harmonic counterpart. For homogeneous background flows, a regularized formulation, analogous to that used in the time-harmonic case, is known to be well-posed in two spatial dimensions [2,4,3]. Similarly as in the time-harmonic case, numerical experiments demonstrate that naive discretizations yield poor approximations [2,4,3]. We note that the system formulation, analogous to formulation (4), has also been studied for general time dependence [17,18]. To the best of our knowledge, well-posedness of such formulation has not been proven.

Derivation of Galbrun's equation from Euler's equations
In this section, we derive Galbrun's equation from Euler's equations via the linearized Euler's equations. We consider an inviscid fluid that either undergoes homentropic flow (the entropy is constant in time and space) or is elastic (the equation of state is independent of the entropy). The time evolution of the flow is assumed to be governed by Euler's equations Dρ + ρ∇ · u = 0, (5b) where u, p, ρ, and ϕ denote the fluid velocity, pressure, density, and volume force density fields, respectively, and D = ∂ t + u · ∇ the material derivative. Equations (5a) and (5b) express conservation of momentum and mass, respectively, while relation (5c) is called the (homentropic) equation of state, which upon differentiation gives the speed of sound c = Σ (ρ).
With the intention to study the evolution of small perturbations of the flow, we introduce the linearization ansatz where φ(x, t) denotes a generic flow field, φ 0 (x, t) is given, and δφ(x, t) denotes the so-called Eulerian perturbation. As before, the given fields u 0 , p 0 , ρ 0 , and ϕ 0 are termed the background flow, and we require that they themselves satisfy Euler's equations, that is, Substituting the linearization ansatz (6) into Euler's equations (5) and retaining terms that are at most linear in the perturbation, we obtain the linearized Euler's equations which describe the evolution of Eulerian perturbations. Informally, the Lagrangian displacement w is the displacement of individual fluid particles, as illustrated in Figure 1. A more precise definition of the Lagrangian displacement as the displacement of individual fluid particles is given in Appendix A. As detailed by Gabard [21], Galbrun's equation may be derived through a Lagrangian linearization of Euler's equations (5), that is, linearizing the equations using the ansatz φ(x, t) = φ 0 (x, t) + δ L φ(x, t), where the Lagrangian perturbation is given by We will, however, derive Galbrun's equation directly from linearized Euler's equations (8). Given δu from linearized Euler's equations (8), we introduce the Lagrangian displacement w abstractly by the relation where L u0 w = (u 0 · ∇)w − (w · ∇)u 0 denotes the Lie derivative of w along u 0 . As will be discussed in the sequel, suitable initial and boundary conditions need to be supplied to equation (10) in order to make w well-defined. That w satisfying relation (10) indeed qualifies as a Lagrangian displacement is motivated in Appendix A. The usefulness of definition (10) stems from the identity which we now demonstrate. By product rule (95), we obtain from definition (10) that Applying the divergence to relation (12) and using identity (94) yields The last term in expression (13) vanishes due to mass conservation (7b), while, using that −ρ −1 0 D 0 ρ 0 = ρ 0 D 0 (ρ −1 0 ), the first two terms may be combined to form the right-hand side of identity (11).
We will now use identity (11) to rewrite equation (8b). To that end, we note that the second and third terms in equation (8b) combine to ∇ · (ρ 0 δu), while the sum of the first and fourth terms can be rewritten as where we in the last step have used mass conservation (7b). Thus, by identity (11) equation (8b) is equivalent to Remark. Note that since δρ + ∇ · (ρ 0 w) = δρ + (w · ∇)ρ 0 + ρ 0 ∇ · w and by definition (9), we obtain from expression (15) that which is the Lagrangian linearization of (5b). (Compare with equation (58) in Gabard [21].) Tentatively assuming that equation (15) implies that and eliminating δu, δρ, and δp from equation (8a), using relations (8c), (10) and (17), we finally attain Galbrun's equation Nevertheless, a lengthy direct calculation yields that formulations (18) and (1) are equivalent. The above presentation deliberately exposes a potential weak link in the derivation of Galbrun's equation, namely the transition from equations (15) to (17). This transition is often referred to as the "no resonance" assumption in the literature-a terminology introduced by Godin [23]-and it will be further analyzed in the sequel.

Preliminaries
Before continuing the investigations into the well-posedness of Galbrun's equation, we briefly recall an abstract framework for time dependent Friedrichs' systems that will be extensively used to assess well-posedness of various initial-boundary value problems that are closely related to Galbrun's equation. The time dependent framework that we rely on was recently presented by Burazin and Erceg [6] and constitutes an extension of a framework for time independent Friedrich's systems presented by Ern et al. [16].
We use the notation from the article by Ern et al. [16]. Let L be a Hilbert space with inner product (·, ·) L and norm · L , and D a dense subspace of L. Moreover, L is identified with its dual L . Let T : D → L andT : D → L be two linear operators that satisfy By W 0 , we denote the completion of D in the inner product (·, ·) L + (T ·, T ·) L (or equivalently in the inner product (·, ·) L + (T ·,T ·) L ). As detailed by Antonić and Burazin [1], the operators T andT can be extended, first by density and then by adjoints, to bounded operators from L to W 0 . Abusing the notation, we still denote these extensions T,T ∈ L(L; W 0 ). The graph space Let V andṼ be subspaces of W that satisfy the conditions The abstract Cauchy problem related to the operator T is given by where the last inequality follows from assumption (T2). Then, the operator A λ0 : where λ 0 is given by relation (21).
Finally, we note that formula (22) yields the estimate which shows that the solution depends continuously on data.

Existence of solutions to Galbrun's equation
In this section, we present a scheme to generate solutions to Galbrun's equation (18) from solutions to linearized Euler's equations (8). The idea is that if δu and δρ are solutions to linearized Euler's equations (8), then the Lagrangian displacement w may be found by solving equation (10). Moreover, provided that the "no resonance" assumption is satisfied, then this w is a solution to Galbrun's equation (18). To validate such a scheme, we investigate • existence of solutions to linearized Euler's equations (8), • existence of solutions to equation (10), and • conditions that guarantee fulfillment of the "no resonance" assumption.

Dissecting the "no resonance" assumption
If we introduce the quantity the "no resonance" assumption takes the form Anticipating that the initial value problem (or depending on the situation, the initial-boundary value problem) that corresponds to the equation to the left of the implication (27) is well-posed, the desired implication would follow if we could provide vanishing data for h. While pursuing this idea, we will reveal that the "no resonance" assumption in some cases imposes a restriction on the Lagrangian displacement w only, and not on the Eulerian perturbations δρ and δu. Assume that Ω ⊂ R d is open, bounded, connected, and lies locally on one side of its Lipschitz boundary ∂Ω. By n we denote the outward unit normal field on ∂Ω. We partition the boundary into three disjoint parts depending on the sign of n · u 0 (28) and we assume that u 0 is such that this partition does not vary with time and dist(Γ − , Γ + ) > 0.
Unless otherwise stated, we assume that the background flow is steady and that the background flow quantities are Lipschitz continuous inΩ. Thus the background flow quantities are bounded in Ω and have bounded first order spatial derivatives almost everywhere inΩ. Moreover, we assume that the density and the speed of sound are bounded away from zero, that is, The initial-boundary value problem related to the "no resonance" assumption reads where 0 < τ < ∞.
Theorem 2. Assume that the background flow is steady. If h I ∈ L 2 (Ω) and h − = 0, then the initial-boundary value problem (30) is well-posed.
Proof. With notation as in Section 3, we let L = L 2 ρ0 (Ω) with inner product (·, ·) L = (ρ 0 ·, ·). By assumption, 0 < infΩ ρ 0 ≤ supΩ ρ 0 < ∞, which implies that L is topologically equivalent to L 2 (Ω). Therefore, in this case, we let C ∞ 0 (Ω) serve as the dense set D ⊂ L described in Section 3. In equation (30a), we find T = u 0 · ∇. By mass conservation (7b) it holds that ∇ · (ρ 0 u 0 ) = 0, which implies that the formal adjoint of T in L isT = −u 0 · ∇ = −T . Thus, T andT satisfy conditions (T1) and (T2), and definition (21) yields λ 0 = 0, since T +T ≡ 0. Due to the assumption that dist(Γ − , Γ + ) > 0, there is a continuous trace operator γ : Remark. Note that a similar estimate to estimate (23) can be derived directly for problem (30), even for unsteady background flows as long as the partitioning (28) does not vary with time. Formally, multiplying equation (30a) by h, integrating over Ω, using integration-by-parts formula (98) and invoking boundary condition (30c), we find that Integrating inequality (31) over the time interval (0, t) and invoking initial condition (30b), we obtain the estimate In either case, Theorem 2 or estimate (32) show that if h I = h − = 0, then the "no resonance" assumption (27) follows. The issue is whether vanishing data for h in problem (30) can be provided for h defined as in expression (26). To that end, assume that δu, δρ satisfy the linearized Euler's equations (8) with suitable initial and boundary conditions, and that the initial datum δρ I of δρ belongs to L 2 (Ω). We start by investigating the initial condition Equation (33) requires that the initial datum w I of the Lagrangian displacement satisfies ∇·(ρ 0 w I ) = −δρ I in Ω, which can be achieved by defining w I = ρ −1 0 ∇v I , where v I ∈ H 1 0 (Ω) is the solution to −∆v I = δρ I in Ω. Thus, using w| t=0 = w I as the initial condition for the Lagrangian displacement, we obtain that h| t=0 = 0. For the case when the background flow is everywhere tangential to ∂Ω-∂Ω = Γ 0 and Γ − = Γ + = ∅-no boundary condition is needed, and the "no resonance" assumption holds if and only if h| t=0 = 0, which can be achieved by adjusting the initial datum of w as demonstrated above. We note that, in this particular case, the "no resonance" assumption imposes no restriction on the initial datum δρ I ∈ L 2 (Ω) of δρ. However, when the background flow is not everywhere tangential to ∂Ω, imposing homogeneous data for h at the boundary part Γ − appears unfortunately to be difficult. Indeed, we would like to impose Contrary to condition (33) that directly translates into a condition on the initial datum of w, it is not possible to convert expression (34) into a condition on the boundary datum of w on Γ − .

Well-posedness of linearized Euler's equations
In this section we prove well-posedness of an initial-boundary value problem for linearized Euler's equations for steady background flows on a bounded domain by employing the framework for abstract Friedrichs' systems that was briefly recalled in Section 3. Following Kreiss and Lorenz [24][Chapter 8.3], we introduce the scaled quantity Then the linearized Euler's equations (8) may be rewritten in the form Similarly as in Section 4.1, we assume that Ω ⊂ R d is open, bounded, connected, and lies locally on one side of its boundary ∂Ω, that the background flow is steady and that the background flow quantities are Lipschitz continuous inΩ, and that the density and the speed of sound are bounded away from zero (29). In addition we assume that ∂Ω is C 1 -regular with a Lipschitz continuous unit normal vector field n, and that the background flow is everywhere tangential to ∂Ω. Recall from Section 4.1 that, since n · u 0 = 0 on ∂Ω, the "no resonance" assumption can be enforced by appropriately choosing the initial datum for the Lagrangian displacement.
Remark. Unless ξ is regular enough, the individual terms in the last two norms might not be well-defined-analogously as the ∂ 1 u 1 term of ∇ · u = ∂ 1 u 1 + ∂ 2 u 2 + ∂ 3 u 3 might not be well-defined for u ∈ H div (Ω). Moreover, since we have assumed that 0 < infΩ ρ 0 ≤ supΩ ρ 0 < ∞, and since B is a bounded operator on L 2 (Ω) d+1 , · W in expression (41) is equivalent to the "standard" The boundary operator D : W → W is given by and we note that for φ, ψ ∈ C 1 (Ω) d+1 , it has the representation where since n · u 0 = 0 on ∂Ω.
We will now proceed to define a trace operator for functions in W that provides a sound mathematical treatment of boundary condition (40c). Our approach is based on the work of Rauch [29], who investigates initial-boundary value problems with vector valued boundary conditions that are characterized using quotient spaces on the boundary. Here, we present a more elementary, nevertheless equivalent, characterization for the scalar valued boundary condition (40c).
Theorem 3. If the boundary is a characteristic surface of constant multiplicity, that is, dim kerA(n) does not vary along the boundary, then Remark. Note that for A(n) defined in expression (44), dim kerA(n) = d − 1, which implies that Theorem 3 is indeed applicable. Moreover, note that functions in C 1 (Ω) d+1 ∩ V satisfy boundary condition (40c) pointwise, while functions in C 1 (Ω) d+1 ∩Ṽ satisfy adjoint boundary condition (47) pointwise.
The following theorem establishes that the spaces V andṼ are "orthogonal" with respect to boundary operator (42).
Proof. By Theorem 3 and its symmetric analogue, it is sufficient to establish the claim for integration-by-parts formula (43) with φ = ξ and ψ =ξ yields where we in the last step employed Lemma 1.

The Lagrangian displacement is well-defined
Here, we make the same assumptions on the domain Ω and on the background flow as in the previous section; in particular, we restrict the attention to the case n · u 0 = 0 on ∂Ω. Employing the abstract framework for Friedrichs' systems, briefly recalled in Section 3, we demonstrate that if we supply an initial condition, the Lagrangian displacement is unambiguously defined by equation (10); that is, the Lagrangian displacement is defined as the solution to the initial value problem where 0 < τ < ∞.
Theorem 7. Assume that n · u 0 = 0 on the boundary. For any initial datum w I ∈ L, the Lagrangian displacement, defined as the solution to initial value problem (57), is well-defined in the sense of Theorem 1.
Recall from Section 4.1 that, since the background flow is assumed to be everywhere tangential to the boundary, we may always choose the initial datum w I in inital condition (57b) so that the "no resonance" assumption is satisfied. Remark. Note that for background flows that cross the domain boundary, we need to supply both an initial condition and a boundary condition on Γ − in order for the Lagrangian displacement to be well-defined by equation (10). Moreover, this general case can be analyzed analogously as in the proof of Theorem 2.

An energy estimate for Galbrun's equation
In this section, we derive an a priori energy estimate for Galbrun's equation (18). In contrast to Section 4 that mostly considers steady background flows, we consider unsteady background flows in this section. As will be seen, the obtained energy estimate for Galbrun's equation has the same form as the one for the first order system which is formed by appending equation (10) to the linearized Euler's equations (36). In equation (58) we have introduced the scaled quantity τ −1 0 w (in units of velocity), where, as before, τ 0 > 0 is an arbitrary time scale. To derive the energy estimates, we assume sufficient regularity of the solution and the background flow. By applying Ω (δu T , δρ, τ −1 0 w T ) to equation (58) from the left and integrating by parts all terms containing first order derivatives (taking advantage of integration-by-parts formula (98)), we obtain where Our first result towards an energy estimate for Galbrun's equation is to show that relation (59) also holds for Galbrun's equation. However, in that case the "no resonance" assumption holds and (17) and definition (35)). Lemma 2. If w is sufficiently regular and satisfies Galbrun's equation (18) for a sufficiently regular background flow, then relation (59) holds with δu = (∂ t + L u0 )w and δρ = −c 0 ρ −1 0 ∇ · (ρ 0 w).
We will once more restrict our attention to the case where the background flow is everywhere tangential to ∂Ω, that is, the case when the "no resonance" assumption can be enforced solely by adjusting the initial datum of the Lagrangian displacement. To derive an energy estimate for Galbrun's equation (18) in this case, we first completely specify the initial-boundary value problem where, as in Section 4.2, Y : ∂Ω → [0, ∞) is a Lipschitz continuous (dimensionless) admittance function with the additional requirement that Y ≥ a > 0, and 0 < τ < ∞. We assume that 0 < ρ 0 := inf sup sup Theorem 8. Assume that w is a sufficiently regular solution to initial boundary value problem (70) for sufficiently regular data and a sufficiently regular background flow that is everywhere tangential to the boundary. For any finite time τ > 0, there exists C, ν > 0 that are independent of w such that for any 0 < t < τ Proof. As in the proof of Lemma 2, we introduce δu and δρ by expressions (61b) and (61c). Moreover, we introduce the corresponding expressions at time t = 0 Then estimate (75) takes the form For convenience and consistent with definition (60), we introduce ξ = (δu, δρ, τ −1 0 w) and ξ I = (δu I , δρ I , τ −1 0 w I ). By the bounds (73) and (74), we obtain from relation (59) that 1 2 where · 2 ρ0 = (ρ 0 ·, ·). We now derive an estimate for the boundary term in expression (78). Observe that by expressions (61b) and (61c), boundary condition (70c) can be reexpressed in δu and δρ, − n · δu + Y δρ = g at ∂Ω for all t ∈ (0, τ ).

Discussion
The abstract definition (10) of the Lagrangian displacement calls for adequate initial and boundary data. In the special case when the background flow is everywhere tangential to the boundary, no boundary datum is needed, and the initial datum can be determined so that the "no resonance" assumption is satisfied. However, when the background flow passes through the boundary of the domain, we do not know whether a boundary datum that guarantees fulfillment of the "no resonance" assumption can be provided for the Lagrangian displacement. We note that fulfillment of the "no resonance" assumption in any case requires that the initial data of δρ and w satisfy condition (33); that is, the "no resonance" assumption imposes a restriction on the initial datum of the Lagrangian displacement. However, at least when the flow is everywhere tangential to the boundary, the "no resonance" assumption imposes no restriction on δu and δρ, that is, it does not restrict the physical behavior of the system.
The role of the "no resonance" assumption has a striking resemblance in electrodynamics. In vacuum, Maxwell's equations are given by where E and B denotes the electric and magnetic fields, J and χ the current and charge densities, µ 0 the vacuum permeability, and 0 the vacuum permittivity [28] where we in expression (84b) have used that ∂ t χ+∇·J = 0, which expresses conservation of electric charge. Thus, if the initial data for E and B satisfy divergence conditions (83c) and (83d), then divergence conditions (83c) and (83d) hold for all subsequent times. Analogously for Galbrun's equation, relation (15) was derived by applying the divergence to definition (10) and invoking equation (8b), which here plays the same role as conservation of charge. If the background flow is everywhere tangential to the boundary of the domain and the initial datum for w satisfies divergence condition (17), then divergence condition (17) holds for all subsequent times. In principle, if the background flow passes through the boundary and w satisfies divergence condition (17) in Ω at t = 0 and on Γ − ⊂ ∂Ω for all t > 0, then divergence condition (17) holds for all subsequent times. As already pointed out above, however, it is not clear how to handle the extra condition on the boundary part Γ − . It would be tempting to define the Lagrangian displacement by both relations (10) and (17), since then the "no resonance" assumption would be automatically satisfied. However, such a definition does not fit the Friedrichs' framework employed here, and at present we do not know how it should be handled. Friedman and Schutz [30] have made some investigations into the matter and remark that if w satisfies both relations (10) and (17), then so does w + ρ −1 0 ∇ × v for any vector field v satisfying ∂ t v + (u 0 · ∇)v + (∇u 0 ) T v = 0. Thus, even when a solution that satisfies both relations (10) and (17) can be found, one may need to deal with a possible non-uniqueness.
We have presented a mildly well-posed initial-boundary value problem for linearized Euler's equations (40) in the special case that the background flow is everywhere tangential to the boundary. Given a solution to that initial-boundary value problem, we may define the Lagrangian displacement by (10) such that relation (17) holds in Ω at t = 0. However, we cannot rigorously conclude that relation (17) holds for all subsequent times and thereby that our Lagrangian displacement satisfies Galbrun's equation (18). The issue is that the derivation of relation (15) appears to require more regularity of δu, δρ and w than what we obtain, at least without performing further analysis. Regularity is also an issue for the energy estimate in Theorem 8 since it hinges on the existence of sufficiently regular solutions to initial-boundary value problem (70). We expect that resolving the regularity issue to be challenging and leave this matter open for future investigation.
Although we have not performed any numerical experiments, the ideas behind our analysis could be transformed into a numerical scheme for solving Galbrun's equation. The resulting scheme would not be computationally attractive since the linearized Euler's equations need to be solved for the Eulerian perturbations before the Lagrangian displacement can be determined from definition (10). To make things worse, it is typically not the Lagrangian displacement but rather the Eulerian perturbations that are the unknowns of interest. Nevertheless, it would be interesting to compare such indirect numerical scheme to other more direct approaches for solving Galbrun's equation or regularized formulations of Galbrun's equation. fluid particle initially located at position p ∈ R 3 can be found by solving the initial value probleṁ X(p, t) = u(X(p, t), t) t > 0 X(p, 0) = p, where the derivative is with respect to time. In the background flow, a fluid particle initially located at position p ∈ R 3 would follow the path X 0 (p, ·) given by the solution of the initial value problemẊ 0 (p, t) = u 0 (X 0 (p, t), t) t > 0 X 0 (p, 0) = p, which is the analogue of problem (85). Recall from Section 2 that u and u 0 are related through u = u 0 + δu, where, as before, δu denotes the Eulerian perturbation of the fluid velocity. For each p and all times t ≥ 0 we define the Lagrangian displacement as the displacement of individual fluid particles W (p, t) = X(p, t) − X 0 (p, t).
We assume that for each t > 0 the mapping X 0 (·, t) is a diffeomorphism and define the vector field w : R d × [0, ∞) → R d such that W (p, t) = w(X 0 (p, t), t).

B Identities for the Lie derivative
In the following, u, v are vector fields and p is a scalar field.