# Mathematical Analysis of Goldstein's Model for time harmonic acoustics in flow

Antoine Bensalah, Patrick Joly, Jean-François Mercier

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Antoine Bensalah, Patrick Joly, Jean-François Mercier. Mathematical Analysis of Goldstein's Model for time harmonic acoustics in flow. ESAIM: Mathematical Modelling and Numerical Analysis, 2022, 56 (2), pp.451-483. 10.1051/m2an/2022007. hal-04832607

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# MATHEMATICAL ANALYSIS OF GOLDSTEIN'S MODEL FOR TIME HARMONIC ACOUSTICS IN FLOW*,**

BENSALAH ANTOINE<fnref n="1" />, JOLY PATRICK<fnref n="2" /> AND MERCIER JEAN-FRANCOIS<fnref n="3" />

**Abstract.** Goldstein's equations have been introduced in 1978 as an alternative model to linearized Euler equations to model acoustic waves in moving fluid. This new model is particularly attractive since it appears as a perturbation a simple scalar model: the potential model. In this work we propose a mathematical analysis of boundary value problems associated with Goldstein's equations in the time harmonic regime.

**Résumé.** Les équations de Goldstein ont été introduites in 1978 comme alternative aux équations d'Euler linéarisées, modélisant la propagation acoustique en écoulement. Ce nouveau modèle est particulièrement attractif car il apparaît comme une perturbation d'un modèle scalaire simple: le modèle potentiel. Dans ce travail, nous proposons une analyse mathématique des problèmes aux limites associés aux équations de Goldstein en régime harmonique.

**1991 Mathematics Subject Classification.** ???, ???

11/07/2021.

## 1. INTRODUCTION

Aeroacoustics concerns the study of the sound propagation in presence of a fluid in flow. In this framework, we aim at determining the propagation of small perturbations of a fluid, namely the acoustic perturbations, created by a known source in an imposed flow [43]. The main motivation is in aeronautics, with the noise reduction of aircraft engines [24]. Applications lie also in the car industry with the need of reducing the sound emitted by exhaust pipes [41, 42, 46], or in the domestic industry with the noise of air-conditioning devices and ventilation ducts.

The most natural model for aeroacoustics is provided by the linearized Euler equations obtained from the linearization of Euler Equations [50] around a stationary mean flow. In what follows, we shall suppose that the fluid is perfect and the flow is homotropic (constant entropy). Euler's

Keywords and phrases: Goldstein's equations, areroacoustics, well-posedness, Fredholm alternative

* ...

** ...

<footnote n="1">¹ antoine.bensalah@airbus.com</footnote>

<footnote n="2">² patrick.joly@inria.fr</footnote>

<footnote n="3">³ jean-francois.mercier@ensta-paris.fr</footnote>
system appears as a first order hyperbolic system with zero order perturbation governing the acoustic velocity $v$ and the acoustic pressure $p$.

Alternative models have been proposed in the literature, motivated by a better fitting to some numerical methods for physical equations. In the mid 1900's, Galbrun's equations have been proposed [27]: the unknown is the so-called Lagrangian displacement field $u$ and the model looks like a vectorial convected wave equation.

In the particular case where the mean flow is potential (i.e. the mean velocity field $v_0$ is the gradient of a scalar potential), one can show [11, 47], under reasonable assumptions about source terms, that the acoustic velocity is itself the gradient of a scalar potential $\varphi$. The model governing $\varphi$, called the potential model in what follows, reduces to a scalar convected wave equation. Because of its simplicity and its adequation to numerical approximations, this scalar model is used in many industrial applications [20, 22], for instance in the analysis of the influence of liners on the acoustic propagation [24, 26, 49].

More recently, Goldstein has proposed a new mathematical model [10, 28]. It can be seen as an extension of the potential one to the general situation where the mean flow is no longer potential. For the applications, the interest is that the corresponding computational code can be built as a modification of existing codes for the potential case. The model couples a scalar potential $\varphi$ to an additional vectorial unknown $\xi$, the hydrodynamic velocity. As we will show in this paper, Goldstein's equations are well adapted for aeroacoustics whereas they are better known in the field of fluid dynamics and where they have been widely used to model the development of perturbations in a swirling flow [2, 18, 19, 29, 30, 33, 51].

Note that both three models can be shown to be equivalent in the case where the mean flow obeys nonlinear stationary Euler equations [50] but each of them can be studied independently. This is important for the development of numerical methods and also for true life applications for which available mean flows are not necessarily exact solutions of stationary Euler equations.

In this article we study Goldstein's equations which did not retain much attention from mathematicians. A particularity of this work is that we are interested in the time harmonic regime: we look for solutions that oscillate in time at a given frequency $\omega > 0$, proportionally to $e^{-i\omega t}$. Our goal is to study the well-posedness (existence and uniqueness of solutions) of the Goldstein's model in this particular case.

There are relatively few mathematical works about aeroacoustic models. In time domain, one can benefit from the well-known theory of symmetric hyperbolic systems in the sense of Friedrichs [25, 37] to prove the well-posedness of the linearized Euler equations. For instance, it has been done in the recent paper [31] where the authors also deduce the well-posedness of Galbrun's equations from the one of Euler's equations.

The analysis of the time harmonic regime appears to be much more delicate. An important assumption, not needed for the time domain analysis, is that the mean flow is **subsonic**, i.e. when
its velocity field has an amplitude strictly less than the speed of sound. This assumption, which is
done in all existing works, is not restrictive with respect to many applications in aeroacoustics.

For the analysis, first order Euler equations appear to be not adapted to a direct mathematical
approach. As a matter of fact, the first existing results concern, to our knowledge, Galbrun's
equations. More precisely the Fredholm nature of the corresponding boundary value problem has
been shown in situations of increasing difficulty. In [12], the case of a 1D shear flow was considered
(one benefits from the simple geometry to use explicit computations). In [13] and [14], the analysis
has been extended to more general 2D mean flows, first in the case of a simplified approximate
model (the so called low Mach model, valid under a smallness assumption about the velocity field
$v_0$), then for the full model. In all cases, the results are obtained under some restrictions about the
variations on the reference mean flow: roughly speaking $|\nabla v_0|$ must be small enough.

The analysis simplifies drastically when one considers the potential model which can be stud-
ied with the same tools and method as the classical Helmholtz equation although the medium is
anisotropic and non homogenous. In [15] (see also [22]), the well-posedness of this model is shown
under the only assumption that the mean flow is subsonic, using Fredholm's alternative.

Very recently, in [32], a work relatively close to ours, the authors study the time harmonic damped
Galbrun's equations in the context of helioseismology. In particular, their model contains absorption
terms that we shall do not consider in this paper. This simplifies in some aspects the analysis which
no longer calls for Fredholm's alternative. The method of analysis is based on some original ad hoc
Helmholtz decomposition of vector fields. The results are obtained under the assumption that the
absorption is large enough but do not require any other assumption on the mean flow, apart its
subsonic nature.

In this paper, we do not consider any intrinsic absorption. It complicates the analysis in particular
through the time harmonic vectorial transport equation satisfied by the unknown $\xi$. The consequence
is that a new restrictive assumption must be done on the reference flow: the $\Omega$-filling condition. In
this work, we prove the existence and uniqueness of solutions to Goldstein's equations for subsonic
$\Omega$-filling flows satisfying an additional condition similar to the one in [14] but more explicit (in
particular easy to check) and this condition can be moreover interpreted as a low vorticity condition.
From the methodological point of view, our method can be seen more as a modification of the
analysis made in [15] for the potential model (this is another advantage of Goldstein's model) and
uses in an essential manner our previous work on the time harmonic transport equation [8] where
the $\Omega$-filling condition, already introduced in [3] for the scalar stationary transport equation, plays
a fundamental role.

The outline of this paper is as follows. In Section 2, we present the problem under consideration,
beginning with the assumptions on the mean flow (Section 2.1), then presenting the governing
equations (Section 2.2) and finally the boundary conditions (Section 2.3). The full problem is
presented in a mathematically oriented manner in Section 2.4. The main section of the paper is
Section 3. Our main results are the object of Section 3.1 in which we present and discuss the
important notion of admissible flow. In Section 3.2, we explain the difficulties of the problem and
present the approach we have chosen. In Section 3.3, we give a recap of the analysis of the potential model. Section 3.4 is devoted to the proof of our main theorem, based on analytic Fredholm theory. Finally, in Section 3.5, we explain in which sense our admissibility condition for the mean flow can be interpreted as a low vorticity condition. The paper is completed by three appendices devoted to a justification of the boundary conditions chosen in Section 2.3 (Appendix A), to the proof of a technical lemma related to Section 3.5 (Appendix B), and to a discussion about a possible alternative approach to the well-posedness analysis (Appendix C).

## 2. EQUATIONS OF THE PROBLEM

### 2.1. Geometry and mean flow

We consider a mean flow occupying $\mathbb{R}^d \setminus \mathcal{O}_R$, $d = 2$ or $3$, where the set $\mathcal{O}_R$ represents a rigid body inside which acoustic waves will not penetrate. The stationary mean flow is characterized by its pressure $p_0$ and velocity vector field $v_0$, all function of the space variable $x$. The constitutive law for a barotropic fluid, namely $p = F(\rho)$, where $F: \mathbb{R} \mapsto \mathbb{R}$ is a smooth non decreasing function, then determines the mean flow density $\rho_0 = F^{-1}(p_0)$ as well as its speed of sound $c_0 > 0$ via $c_0^2 := F'(\rho_0)$ [50]. The quantities $(\rho_0, p_0, v_0)$ are supposed to satisfy stationary Euler equations [50] (see however Remark 1) and in particular the mass conservation:

$$ \mathrm{div}(\rho_0 v_0) = 0, \qquad (1) $$

which is the only equation on the mean flow that we shall use explicitly in this paper.

**Remark 1.** Real life computations in aeroacoustics are often done with idealized mean flows that are not necessary exact solutions of stationary Euler equations but result of various approximations (constant density for instance) that may due to physical simplifications or due to approximate numerical calculations. For this paper, the mass conservation condition (1) is the only equation that we shall use explicitly.

For the application to acoustics, we assume that the velocity field is smooth enough, $v_0 \in C^1(\mathbb{R}^d \setminus \mathcal{O}_R; \mathbb{R}^d)$ and that, as $\mathcal{O}_R$ is rigid, the flow is sliding along $\partial\mathcal{O}_R$, that is to say, $n(\mathbf{x})$ denoting the unit normal on $\partial\mathcal{O}_R$,

$$ \forall \mathbf{x} \in \partial\mathcal{O}_R, \quad v_0(\mathbf{x}) \cdot n(\mathbf{x}) = 0. $$

We also assume that the density $\rho_0$ and the velocity $c_0$ are bounded from below and above by two strictly positive constants:

$$ \forall \mathbf{x} \in \mathbb{R}^d \setminus \mathcal{O}_R, \quad 0 < \rho_- \le \rho_0(\mathbf{x}) \le \rho_+, \quad 0 < c_- \le c_0(\mathbf{x}) \le c_+. \qquad (2) $$

We are interested to study the propagation of acoustic waves in a connected and bounded domain $\Omega \subset \mathbb{R}^d \setminus \mathcal{O}_R$ whose boundary $\partial\Omega$ is split into two parts:

$$ \partial\Omega = \Gamma \cup \Gamma_R, $$

where $\Gamma_R := \partial\Omega \cap \partial\mathcal{O}_R$ is the rigid part and $\Gamma$ is the outer boundary. It will also be useful, for formulating boundary conditions, to separate $\Gamma$ into three parts:

$$ \Gamma = \Gamma_+ \cup \Gamma_- \cup \Gamma_0, $$
where, by definition

$$
\begin{align}
\Gamma_+ &:= \{\boldsymbol{x} \in \Gamma, \boldsymbol{n}(\boldsymbol{x}) \cdot \boldsymbol{v}_0(\boldsymbol{x}) > 0\} && \text{is the outflow boundary,} \nonumber \\
\Gamma_- &:= \{\boldsymbol{x} \in \Gamma, \boldsymbol{n}(\boldsymbol{x}) \cdot \boldsymbol{v}_0(\boldsymbol{x}) < 0\} && \text{is the inflow boundary,} \tag{3} \\
\Gamma_0 &:= \{\boldsymbol{x} \in \Gamma, \boldsymbol{n}(\boldsymbol{x}) \cdot \boldsymbol{v}_0(\boldsymbol{x}) = 0\} && \text{is the sliding boundary.} \nonumber
\end{align}
$$

For the mathematical analysis, we shall assume that the inflow and outflow boundaries are well separated, namely (such a condition appears, for instance, in most mathematical works about the stationary transport equation)

$$d(\Gamma_{-}, \Gamma_{+}) > 0. \tag{4}$$

**Remark 2.** As seen in (3), the boundaries are defined by the flow so that they should be denoted $\Gamma_{\pm}(\boldsymbol{v}_0)$ and $\Gamma_0(\boldsymbol{v}_0)$. We did not do so for avoiding heavy notation.

To illustrate our purpose, let us consider two examples of “real life” applications.

**Application 1:** Propagation of acoustic waves in a deformed duct.

Denoting $\boldsymbol{x} = (\boldsymbol{x}_T, x_d)$, with $\boldsymbol{x}_T \in \mathbb{R}^{d-1}$ the transverse variable and $x_d \in \mathbb{R}$ the longitudinal one, we consider that the fluid domain $\mathbb{R}^d \setminus \mathcal{O}_R$ is an infinite “deformed cylinder”, i.e. a infinite connected domain that is transversally bounded

$$\exists R_0 > 0 \quad \text{s.t.} \quad \mathbb{R}^d \setminus \mathcal{O}_R \subset \{\boldsymbol{x} \in \mathbb{R}^d / |\boldsymbol{x}_T| < R_0\},$$

and, outside a bounded set, is perfectly cylindrical, namely $S_{\pm}$ denoting two bounded domains of $\mathbb{R}^{d-1}$

$$\exists L > 0 \quad \text{s.t.} \quad \mathbb{R}^d \setminus \mathcal{O}_R \cap \{\pm x_d > L\} = (S_{\pm} \times \mathbb{R}) \cap \{\pm x_d > L\}.$$

We also assume that the flow is homogeneous outside a bounded domain: there exists positive constants $\rho_{\infty}^{\pm}, c_{\infty}^{\pm}$ and $v_{\infty}^{\pm}$ such that, $e_d$ being the unit vector in the direction $x_d$,

$$\pm x_d > L \quad \implies \quad \rho_0(\boldsymbol{x}) = \rho_{\infty}^{\pm}, \quad c_0(\boldsymbol{x}) = c_{\infty}^{\pm} \quad \text{and} \quad \boldsymbol{v}_0(\boldsymbol{x}) = v_{\infty}^{\pm} e_d.$$

In this case, the domain of interest for the propagation of acoustic waves will be typically

$$\Omega := \{\boldsymbol{x} \in \mathbb{R}^d \setminus \mathcal{O}_R / |x_d| < L\},$$

whose outer boundary $\Gamma$ is such that $\Gamma_- = S_- \times \{-L\}$ and $\Gamma_+ = S_+ \times \{L\}$.

**Application 2:** A model problem in aeronautics ($d = 3$).

This concerns a more industrial application about modeling the noise produced by airplanes during their flight. In cruise regime, the airplane moves at a constant speed $v_{\infty} e_d$ and, for the modeling, the idea is to stand in the attached moving frame. In this way, everything happens as if the plane, which will be typically the rigid body $\mathcal{O}_R$, created a mean flow whose velocity would be constant “at infinity”, which means sufficiently far, equal to $v_{\infty} e_d$. This flow is typically obtained from a CFD calculation solving stationary Euler (or Navier-Stokes) equations. More precisely, one
generally assumes that there exists a (sufficiently large) parallelepipedic box $B$ (which will contain the computational domain), outside which the mean flow is supposed to be homogeneous, namely

$$ \forall \mathbf{x} \in \mathbb{R}^d \setminus \Omega, \quad \rho_0(\mathbf{x}) = \rho_\infty, \quad c_0(\mathbf{x}) = c_\infty \quad \text{and} \quad v_0(\mathbf{x}) = v_\infty e_d. $$

In that case, the computational domain is $\Omega = B \setminus O_R$ and the outer boundary $\Gamma$ is $\partial B$. The boundary $\Gamma_0$ is the union of the four faces of $B$ that are parallel to $e_d$, the inflow boundary $\Gamma_-$ is the face of $B$ that has $e_d$ as incoming normal vector and the outflow boundary $\Gamma_+$ is the face of $B$ that has $e_d$ as outgoing normal vector (see Fig. 1).

<caption>FIGURE 1. Typical mean flows. Application 1 (right). Application 2 (left)</caption>

## 2.2. Time harmonic Goldstein's equations

The goal of the modeling of acoustics in a stationary mean flow (characterized by $(v_0, p_0)$) is to compute the perturbations $(v(x, t), p(x, t))$ induced by a small perturbative source term. If this source term varies in time proportionally to $e^{-i\omega t}$, for a given frequency $\omega > 0$ and if we look for the first order term of the perturbation (with respect to the small amplitude of the source term), it is natural to look for acoustic perturbations of the form $v(x) e^{-i\omega t}$ for the velocity and $p(x) e^{-i\omega t}$ for the pressure where $(v, p)$ are complex valued functions. The unknowns $(v, p)$ are naturally governed by time harmonic linearized Euler equations [50]. However, the Goldstein's model is better adapted for taking into account the fact that the nature of the acoustic perturbations depends on the characteristics of the flow and in particular on its vorticity $\omega_0 := \nabla \times v_0$ (see also Remark 3).

When the flow is potential ($\omega_0 = 0$) and homotropic (constant entropy), $v$ is found to be potential, i.e. $v = \nabla\varphi$, where the velocity potential $\varphi$ satisfies the convected Helmholtz equation [11, 47], in which the acoustic source is represented by the right hand side $f$:

$$ D_{\omega} (c_{0}^{-2} D_{\omega} \varphi) - \rho_{0}^{-1} \mathrm{div} (\rho_{0} \nabla \varphi) = f. \quad (5) $$

In this equation, $D_\omega := -i\omega + v_0 \cdot \nabla$, is the harmonic convective derivative. (5) is the form of the convected wave equation that is the most commonly used in the literature [11, 15]. However, exploiting the mass conservation equation (1), this equation can be rewritten in divergence form, more suitable for the mathematical and numerical analysis. This exploits the fact that, for any scalar function $\psi$,

$$ \rho_0 D_\omega \psi = -i\omega \rho_0 \psi + \rho_0 v_0 \cdot \nabla \psi = -i\omega \rho \psi + \mathrm{div}(\rho_0 v_0 \psi). $$

Applying the above with $\psi = c_0^{-2} D_\omega \phi$, (5) (multiplied by $\rho_0$) can be rewritten as

$$ -\mathrm{div} (\rho_0 (\nabla \varphi - c_0^{-2} D_\omega \varphi v_0)) - i\omega \rho_0 c_0^{-2} D_\omega \varphi = \rho_0 f. \quad (6) $$

For a general flow of vorticity $\omega_0 \neq 0$, the acoustic perturbations are also found vortical. That is why, in addition to the potential $\varphi$, one has to introduce a new (vector valued) unknown: the
hydrodynamic vector field $\xi$. These unknowns are found to satisfy the Goldstein equations [10, 28], that we write below in divergence form, in conformity with (6):

$$ egin{cases} -\operatorname{div}(\rho_0(\nabla\varphi + \xi - c_0^{-2}D_\omega\varphi v_0)) - i\omega\rho_0 c_0^{-2}D_\omega\varphi = \rho_0 f, & (i) \\ D_\omega\xi + (\xi \cdot \nabla)v_0 = \nabla\varphi \times \omega_0, & (ii) \end{cases} \quad (7) $$

and are linked to the Euler's unknowns, velocity $v$ and pressure $p$, by

$$ egin{cases} v = \nabla\varphi + \xi, & (i) \\ p = -\rho_0 D_\omega\varphi. & (ii) \end{cases} \quad (8) $$

Let us interpret each equation. First (7)(i) means that, given $\xi$, $\varphi$ is solution of the convected Helmholtz equation, with source term $f + \rho_0^{-1} \operatorname{div}(\rho_0 \xi)$. Next (7)(ii) means that, given $\varphi$, $\xi$ is solution of a time harmonic transport equation, namely of the form

$$ \mathcal{T}_0(\omega)\xi = g, \quad \text{with} \quad \mathcal{T}_0(\omega)\xi := D_\omega\xi + (\xi \cdot \nabla)v_0, \quad (9) $$

by definition the time harmonic transport operator, with source term $g = \nabla\varphi \times \omega_0$. Of course, (7) needs to be completed with boundary conditions (object of the next section).

**Remark 3.** Another drawback of time harmonic linearized Euler equations is that, contrary to what happens in time domain, there are not well established numerical methods for their resolution. This is not the case of the Goldstein's model.

## 2.3. Boundary conditions for the perturbations

As we work with a system of two equations (7), we need two different boundary conditions. The first one, called the acoustic condition, will be seen as a boundary condition for the potential $\varphi$ and naturally attached to (7)(i) while the second one, called the hydrodynamic condition, will be seen as a boundary condition for the hydrodynamic velocity $\xi$ and naturally attached to (7)(ii). However, in fact, as it is the case of the two equations in (7), these conditions (the acoustic condition to be more precise) couple the two unknowns.

### 2.3.1. Acoustic condition

This condition writes differently depending on the part of the boundary, $\Gamma_R$ or $\Gamma$, one is looking at.

* On the rigid boundary $\Gamma_R$, the boundary condition to be chosen is clear: as for the mean flow, the velocity of the acoustic disturbances is tangential. According to (8)(i), this condition, namely $v \cdot n = 0$, simply reads:

$$ (\nabla\varphi + \xi) \cdot n = 0 \quad \text{on} \quad \Gamma_R. \quad (10) $$

* On the artificial boundary $\Gamma$, the choice of good boundary conditions, which are supposed to represent the fact that acoustic waves want to leave the domain $\Omega$ through $\Gamma$, is a delicate issue: this is where approximate modeling enters into account. In the context of this paper, we will content ourselves with a first order absorbing boundary condition (or first order
radiation condition in the spirit of [23], see also Remark 4 for more accurate and sophisticated alternatives). This condition writes

$$ (\nabla\varphi + \xi) \cdot n - (v_0 \cdot n) c_0^{-1} (\xi \cdot n + c_0^{-1} D_\omega \varphi) = i \frac{\omega}{c_0} \varphi \quad \text{on } \Gamma, \qquad (11) $$

and its derivation is explained in Appendix A: as for the standard Helmholtz equation, it is designed in order to perfectly absorb the waves that strike the artificial boundary $\Gamma$ with normal incidence. By the way, in the absence of flow (i.e. $v_0$ and $\xi$ vanishing everywhere), one recovers the well known first order (or Sommerfeld) absorbing condition $\nabla\varphi \cdot n = i \frac{\omega}{c_0} \varphi$.

The reader will note that, as $v_0 \cdot n = 0$ on $\Gamma_R$, introducing the function $\lambda : \partial\Omega \to \mathbb{R}^+$ such that $\lambda = 0$ on $\Gamma_R$ and $\lambda = 1$ on $\Gamma$ the conditions (10) and (11) can be gathered in the following unified form,

$$ \rho_0(\nabla\varphi + \xi - c_0^{-2} D_\omega \varphi v_0) \cdot n - \rho_0 c_0^{-1} (v_0 \cdot n)(\xi \cdot n) = i \lambda \frac{\omega}{c_0} \rho_0 \varphi \quad \text{on } \partial\Omega, \qquad (12) $$

which is, thanks to the multiplication by $\rho_0$, compatible with the divergence form of the equation (7)(i). In the rest of the paper, we will consider the boundary condition (12) in the more general case where $\lambda$ is an impedance function along $\partial\Omega$ satisfying

$$ \lambda : \partial\Omega \to \mathbb{R}^+ \in L^\infty(\Gamma) / \exists \gamma \subset \partial\Omega \text{ s.t. meas}_{\partial\Omega} \gamma > 0 \text{ and } \lambda > 0 \text{ on } \gamma. \qquad (13) $$

In particular, this includes the case of a boundary partially made of an absorbing wall with impedance $\lambda$.

**Remark 4.** There are several approaches to define alternative radiation conditions which are more efficient than the first order that we used. Let us quote two of them.

* In the situation described in **Application 2**, Section 2.1, where the mean flow is supposed to be homogeneous at the outer boundary, one can construct transparent boundary conditions via an integral representation formula [15]. Such a condition is thus non local. The corresponding boundary value problem has been analyzed in [15] in the case where the flow is potential everywhere (so that $\varphi$ is the only unknown) and at rest outside $\Omega$.

* An alternative to radiation conditions is provided by Perfectly Matched Layers (PMLs): the absorption of waves is realized inside an absorbing layer instead across the absorbing boundary (see [9, 39] for introductory papers). There is not much analysis of PMLs for acoustics in flow. It is nevertheless worthwhile mentioning the works [6] for the convected Helmholtz equation or [5] for Galbrun's equations, in the case of a waveguide (a situation that enters the framework of **Application 1**, Section 2.1).

### 2.3.2. Hydrodynamic condition

Because of the nature of the equation (14)(ii) satisfied by $\xi$, transport type equation, we need a boundary condition only on the inflow boundary $\Gamma_-$. We suppose that this boundary is located in such a way that there does not exist any acoustic source upstream and that the vorticity of the mean flow vanishes in the upstream area. As a consequence, we prescribe that $\xi$ vanishes on $\Gamma_-$ (this can be interpreted as a causal boundary condition)

$$ \xi = 0 \quad \text{on } \Gamma_- $$
## 2.4. Mathematical formulation of the boundary value problem

We now describe the mathematical problem to be solved, making precise in which functional spaces the unknowns are searched and the data are taken:

Given $f \in L^2(\Omega)$, find $(\varphi, \boldsymbol{\xi}) \in H^1(\Omega) \times L^2(\Omega)^d$, such that

$$ egin{cases} -\operatorname{div}(\rho_0(\nabla\varphi + \boldsymbol{\xi} - c_0^{-2}D_\omega\varphi\mathbf{v}_0)) - i\omega\rho_0 c_0^{-2}D_\omega\varphi = \rho_0 f, & \text{in } \Omega, \quad (i) \\ D_\omega\boldsymbol{\xi} + (\boldsymbol{\xi} \cdot \nabla)\mathbf{v}_0 = \nabla\varphi \times \omega_0, & \text{in } \Omega, \quad (ii) \end{cases} $$

with the following acoustic and hydrodynamic boundary conditions

$$ egin{cases} \rho_0(\nabla\varphi + \boldsymbol{\xi} - c_0^{-2}D_\omega\varphi\mathbf{v}_0) \cdot \mathbf{n} - \rho_0 c_0^{-1}(\mathbf{v}_0 \cdot \mathbf{n})(\boldsymbol{\xi} \cdot \mathbf{n}) = i\lambda c_0^{-1} \rho_0 \varphi, & \text{on } \partial\Omega, \quad (i) \\ \boldsymbol{\xi} = 0, & \text{on } \Gamma_{-}. \end{cases} $$

Let us comment the physical pertinence of the functional framework and the sense to give to the boundary conditions (15), which is related to the existence of appropriate traces. First note that looking for $\varphi \in H^1(\Omega)$ and $\boldsymbol{\xi} \in L^2(\Omega)^d$ implies, via $\mathbf{v} = \nabla\varphi + \boldsymbol{\xi}$, that

$$ \int_{\Omega} \rho_0 |\mathbf{v}|^2 < +\infty, $$

namely that the solution has a finite kinetic energy.

The question of traces is a little bit more delicate. First, we remark that (14)(i) implies

$$ \rho_0(\nabla\varphi + \boldsymbol{\xi} - c_0^{-2}D_\omega\varphi\mathbf{v}_0) \in H(\operatorname{div}; \Omega), $$

so that the usual trace theorem in $H(\operatorname{div}; \Omega)$ ensures that

$$ \rho_0(\nabla\varphi + \boldsymbol{\xi} - c_0^{-2}D_\omega\varphi\mathbf{v}_0) \cdot \mathbf{n} \in H^{-\frac{1}{2}}(\partial\Omega). \quad (16) $$

Note also that (14)(ii) implies that $\boldsymbol{\xi}$ belongs to an anisotropic Sobolev space, namely

$$ \boldsymbol{\xi} \in H(\Omega, \mathbf{v}_0) := \{\boldsymbol{\zeta} \in L^2(\Omega)^d / (\mathbf{v}_0 \cdot \nabla)\boldsymbol{\zeta} \in L^2(\Omega)^d\}, $$

where $H(\Omega, \mathbf{v}_0)$ is an Hilbert space for the natural graph norm

$$ \|\boldsymbol{\zeta}\|_{H(\Omega, \mathbf{v}_0)}^2 := \int_{\Omega} (|\boldsymbol{\zeta}|^2 + |(\mathbf{v}_0 \cdot \nabla)\boldsymbol{\zeta}|^2). $$

Next, it is well known [21, 38] that, under the separation condition (4), the trace $\boldsymbol{\zeta}|_{\Gamma_\pm}$ on $\Gamma_\pm$ of any $\boldsymbol{\zeta} \in H(\Omega, \mathbf{v}_0)$ is well defined and that

$$ \boldsymbol{\zeta}|_{\Gamma_{\pm}} \in L^2(\Gamma_{\pm}, |\mathbf{v}_0 \cdot \mathbf{n}|)^d := \{\boldsymbol{\zeta} : \Gamma_{\pm} \to \mathbb{C}^d / \int_{\Gamma_{\pm}} |\mathbf{v}_0 \cdot \mathbf{n}| |\boldsymbol{\zeta}|^2 < +\infty\}. \quad (17) $$

Moreover, the trace map $\boldsymbol{\zeta} \mapsto \boldsymbol{\zeta}|_{\Gamma_\pm}$ is continuous from $H(\Omega, \mathbf{v}_0)$ in $L^2(\Gamma_\pm, |\mathbf{v}_0 \cdot \mathbf{n}|)^d$. This gives a sense to (15)(ii) which can also be rewritten as

$$ \boldsymbol{\xi} \in H^{-}(\Omega, \mathbf{v}_0) := \{\boldsymbol{\zeta} \in H(\Omega, \mathbf{v}_0) / \boldsymbol{\zeta} = 0 \text{ on } \Gamma_{-}\} \quad (\text{closed in } H(\Omega, \mathbf{v}_0)). $$
This also gives a sense to the trace $(\zeta \cdot n)(v_0 \cdot n)$, when $\zeta \in H^{-}(\Omega, v_0)$, as an element of $L^2(\partial\Omega)$, that vanishes along $\Gamma_0 \cup \Gamma_R$ (because $v_0 \cdot n = 0$) and $\Gamma_-$ (because $\xi = 0$). Indeed, it suffices to check that $(\zeta \cdot n)(v_0 \cdot n) \in L^2(\Gamma_+)$ which follows from the following estimate

$$ \int_{\Gamma_+} |(\zeta \cdot n)(v_0 \cdot n)|^2 \leq \int_{\Gamma_+} |\zeta \cdot n|^2 |v_0 \cdot n|^2 \leq \|v_0\|_{L^\infty(\Omega)} \int_{\Gamma_+} |\zeta \cdot n|^2 |v_0 \cdot n| $$

which is finite thanks to (17). Then, by continuity of the trace operator $\zeta \mapsto \zeta|_{\Gamma_+}$, we know that there exists a constant $C_+ > 0$, that depends only on $\Omega$, $\Gamma_+$ and $v_0$ such that

$$ \forall \zeta \in H^{-}(\Omega, v_0), \quad \|(\zeta \cdot n)(v_0 \cdot n)\|_{L^2(\Gamma_+)} \leq C_+ \|\zeta\|_{H(\Omega, v_0)}. \quad (18) $$

Finally (16) and (18) give a sense to the boundary condition (15)(i) in $H^{-\frac{1}{2}}(\partial\Omega)$ a priori. However, since $\varphi|_{\partial\Omega} \in H^{\frac{1}{2}}(\partial\Omega) \subset L^2(\partial\Omega)$ and $\lambda \in L^\infty(\partial\Omega)$, $\lambda \rho_0 c_0^{-1} \varphi \in L^2(\partial\Omega)$ so that (15)(i) implies that the trace (16) belongs to $L^2(\partial\Omega)$ and (15)(i) holds in $L^2(\partial\Omega)$.

### 3. ANALYSIS OF GOLDSTEIN'S EQUATIONS

#### 3.1. The main result

Our main result will rely on a particular assumption of the flow $v_0$ with respect to the domain $\Omega$. The first important condition is related to the following definition, that we choose first to express in “physical” terms.

**Definition 1** ($\Omega$-filling flow – informal). The vector field $v_0$ is said $\Omega$-filling if there exists a time upper bound $t^* > 0$ such that any point inside $\Omega$ is reached before $t^*$ by a particle that is transported by the flow from a point on the inflow boundary $\Gamma_-$ at time $t = 0$.

**Remark 5.** The $\Omega$-filling condition excludes in particular two situations:

* the existence of closed streamlines (also called recirculations or periodic orbits) for the flow $v_0$ inside $\Omega$. These would correspond to the existence of particles that are transported by the flow and stay indefinitely inside $\Omega$.

* the existence of stopping points, i.e. points $x_s \in \bar{\Omega}$ where $v_0(x_s) = 0$: this corresponds to the existence of particles that take an arbitrarily large time to reach a point arbitrarily closed to $x_s$.

In dimension $2$, $d = 2$, there exists a particularly simple characterization of $\Omega$-filling flows provided that $\Omega$ is simply connected. The result, proven in [8], is linked to Brouwer and Poincaré-Bendixson theorems [17], exploits the fact that, roughly speaking, the existence of periodic orbits implies the existence of a stopping point. The precise statement is the following

$$ \text{If } \Omega \subset \mathbb{R}^2 \text{ is simply connected, } v_0 \text{ is } \Omega\text{-filling} \iff \inf_{x \in \Omega} |v_0(x)| > 0. \quad (19) $$

**Remark 6.** Obviously, if the statement of Definition 1 is satisfied for some $t^* > 0$, it remains true for any larger time. In the sequel, we shall denote $t^*(v_0, \Omega)$ the infimum of such times and we shall call it the life time of the flow in $\Omega$.
The Ω-filling condition in Definition 1 will be stated in a more mathematical form when we shall use it, in Section 3.4.2. We shall need a more elaborate condition that relies on the introduction of the following functions

$$ \Psi(s) := \sqrt{\frac{e^s - 1}{s}}, \quad \Phi(s) := \frac{\sqrt{e^s - 1 - s}}{|s|}, \quad s \in \mathbb{R}^*, \quad \Psi(0) = 1, \quad \Phi(0) = 1/\sqrt{2}, \quad (20) $$

whose useful properties are summarized in the following lemma

**Lemma 1.** The functions $\Psi$ and $\Phi$ are continuous and strictly increasing from $0$ ($s \to -\infty$) to $+\infty$ ($s \to +\infty$).

**Proof.** The proof is easy but given for completeness as the result is important for the rest of the paper.

The function $f(s) := \Psi(s)^2$ is analytic on $\mathbb{R}$. Given $s \in \mathbb{R}^*$, $f'(s) = h(s)/s^2$ where $h(s) = e^s(s - 1) + 1$. Providing $h'(s) = se^s$, $h$ is decreasing on $\mathbb{R}_-$ and increasing on $\mathbb{R}_+$, and thanks to $h(0) = 0$, one deduces that $f'(s) = h(s)/s^2$ is positive on $\mathbb{R}^*$ and one computes $f'(0) = 1/2$. Thus, $f = \Psi^2$ is strictly increasing, so is $\Psi$.

In the same fashion, for $\Phi$, the function $g(s) := \Phi(s)^2$ is analytic on $\mathbb{R}$. Given $s \in \mathbb{R}^*$, $g'(s) = H(s)/s^3$ where $H(s) = e^s(s - 2) + s + 2$ of derivative $H'(s) = h(s)$. We have just seen that $h(s) > 0$ on $\mathbb{R}^*$ and $h(0) = 0$, then, $H$ is increasing on $\mathbb{R}$ and from $H(0) = 0$, one deduces that $g'(s) = H(s)/s^3$ is positive on $\mathbb{R}^*$ and thus $g = \Phi^2$ is strictly increasing and so is $\Phi$. $\square$

**Definition 2** (Admissible flow). An flow $v_0$ is admissible if and only if it is Ω-filling, with life time $t^*(v_0, \Omega)$, and denoting $\omega_0 := \nabla \times v_0$ (its vorticity) and $M_0 := v_0/c_0$ (its Mach number), satisfies the following inequality

$$ \|\omega_0\|_{L^\infty} t^*(v_0, \Omega) \Phi(2t^*(v_0, \Omega) \|\nabla v_0\|_{L^\infty}) < 1 - \|M_0\|_{L^\infty}^2. \quad (21) $$

At first glance, (21) appears as an upper bound for the $L^\infty$ norm of the vorticity $\omega_0$. This is at second glance not so obvious since the life time $t^*(v_0, \Omega)$ depends on $v_0$ in a complicated implicit manner while $\omega_0$ is also related to $v_0$. To emphasize that (21) does indeed correspond to a smallness of the vorticity, we refer the reader to Section 3.5.

**Theorem 1.** Assume that $v_0$ is an admissible flow in the sense of Definition 2 and that the impedance function $\lambda$ satisfies (13) and that the inflow and outflow boundaries are well separated, i.e. (4). Then, there exists a discrete set $\mathcal{R}_{ex} \subset \mathbb{R}$ of exceptional frequencies such that, for any $\omega \notin \mathcal{R}_{ex}$, the boundary value problem (14, 15) is well-posed. More precisely, for such a frequency and for any $f \in L^2(\Omega)$, (14, 15) has a unique solution $(\varphi, \xi) \in H^1(\Omega) \times L^2(\Omega)^d$ that depends continuously on $f$: for some constant $C(v_0, \Omega)$

$$ \|\varphi\|_{H^1(\Omega)} + \|\xi\|_{L^2(\Omega)^d} \le C(v_0, \Omega) \|f\|_{L^2(\Omega)}. $$

Any frequency in the set $\mathcal{R}_{ex}$ would correspond to a resonance, a frequency for which there exists a non zero finite energy to the homogeneous problem. For a time source with this frequency, the solution of the evolution problem would blow up when $t \to +\infty$ instead of "converging" to a time harmonic solution (limiting amplitude principle). The existence of such frequencies - which cannot
be excluded a priori because of our method of proof - remains an open question (see also the point (d) in the discussion below).

It is interesting to question the importance / relevance of the assumptions of theorem 1:

(a) The Ω-filling condition, which is implicitly included in the admissibility condition, will be used in our analysis to eliminate the unknown $\xi$ (cf. Section 3.4.2). Violating this condition disqualifies our approach but is not a priori an obstacle to the well-posedness of (14, 15). However, it has been shown in the PhD thesis [7] that, in some specific situations, the existence of closed orbits (closed streamlines) was, for a wide range of frequencies, an obstacle to the well-posedness of (14, 15), at least in the framework adopted in Section 2.4. For such frequencies, it would be necessary to adopt a new notion of weak solution that would require to accept more singular solutions than $(\varphi, \xi) \in H^1(\Omega) \times L^2(\Omega)$. Similar phenomena appear for the propagation of electromagnetic waves in magnetized plasmas [45].

(b) The separation condition (4) appears regularly in the theory of transport equations (see for instance [21]). Clearly for us, it will appear again as a technical condition for eliminating $\xi$. However, contrary to the Ω-filling condition, our feeling is that this condition is not really essential and could be removed.

(c) The admissibility condition of Definition 2 is stronger than simply saying that the mean flow is strictly subsonic, $||M_0||_{L^\infty} < 1$, a condition that is made in most mathematical studies in aeroacoustics. If the flow became supersonic in parts of $\Omega$, the nature of the equation (14) governing $\varphi$ would change since the ellipticity of the principal part of the differential operator would be lost. Then, the mathematical analysis would fall under the application of completely different techniques, far beyond the domain of competence of the authors.

(d) The admissibility condition will be used in our approach to show that the reduced problem falls under Fredholm's alternative. In this paper, this condition will clearly appear as a technical condition not connected to physics. However, it is not so surprising to see appear a condition of this nature, which corresponds to imposing an upper bound on the velocity field $v_0$ space variations, and more precisely an upper bound on the vorticity $\omega_0$. Indeed, it is well-known, in particular in the case of a laminar flow inside a waveguide, that a too strong vorticity is the cause of the development of hydrodynamic instabilities, in particular instabilities of Kelvin-Helmholtz type [44] which questions the soundness of the time harmonic model [1]. This is clearly very close to the question of the existence of resonances as evoked previously.

## 3.2. Orientation and difficulties

The method we shall follow is to see the coupled problem (14,15) as a perturbation of the problem obtained by taking $\omega_0 = 0$ in (14,15), which is the problem to be solved in the case of a potential flow, i.e. when $\nabla \times v_0 = 0$. Assume for a while that, given $\varphi$, the transport equation (14)(ii) in $\xi$, completed by the boundary condition (15)(ii), is well-posed. In such a case, one easily infers that, since the source term $\nabla\varphi \times \omega_0$ vanishes, $\xi = 0$. As a consequence, the only unknown is the potential $\varphi$ that is solution of the convected Helmholtz problem

$$ egin{cases} -\operatorname{div}(\rho_0(\nabla\varphi - c_0^{-2}D_\omega\varphi v_0)) - i\omega\rho_0 c_0^{-2}D_\omega\varphi = \rho_0 f, & \text{in } \Omega, \quad (i) \\ \rho_0(\nabla\varphi - c_0^{-2}D_\omega\varphi v_0) \cdot n = i\lambda c_0^{-1}\rho_0\varphi & \text{on } \partial\Omega. \quad (ii) \end{cases} \tag{22} $$
In the context of subsonic flows to which we restrict ourselves in this paper, the analysis on the above problem is well known (see for instance [15]) but for the sake of completeness and pedagogy, we shall recall the main ideas and results in Section 3.3.

To solve the Goldstein's problem (14,15), we shall use a perturbative analysis. The idea is to eliminate $\xi$ via the solution of the transport problem ((14)(ii),(15)(ii)). This, at least formally for the moment (this will be made precise and rigorous in Section 3.4.2, see (47), (48)), allows us to express $\xi$ as a function of $\varphi$

$$ \xi = S_0(\omega; \nabla \varphi). \qquad (23) $$

As already seen in Section 3.1, a sufficient condition (and probably necessary) for the solvability of (14)(ii) is that the flow is $\Omega$-filling (Definition 1). This was demonstrated in [8] and will be recalled (and extended) in Section 3.4.2. Thanks to (23), we are led to a "modified" convected Helmholtz problem in $\varphi$

$$ \begin{cases} -\operatorname{div} (\rho_0(\nabla\varphi + S_0(\omega; \nabla\varphi) - c_0^{-2}D_\omega\varphi v_0)) - i\omega\rho_0 c_0^{-2}D_\omega\varphi = \rho_0 f, & \text{in } \Omega, \quad (i) \\ \rho_0(\nabla\varphi + S_0(\omega; \nabla\varphi) - c_0^{-2}D_\omega\varphi v_0) \cdot n - \rho_0 c_0^{-1}(v_0 \cdot n)(S_0(\omega; \nabla\varphi) \cdot n) = i\lambda c_0^{-1}\rho_0 \varphi, & \text{on } \partial\Omega, \quad (ii) \end{cases} \qquad (24) $$

that we see as a perturbation of (22), reason why the idea to solve (24) by adapting the arguments used for solving (22). We shall also call the problem (24) the reduced Goldstein's problem in the sense that the only unknown is the potential $\varphi$.

This is in fact not as easy that one might think. As we shall recall in Section 3.3, the analysis of the convected Helmholtz equation is very close to the one of the standard Helmholtz equation. One first reduces the problem to the application of Fredholm's alternative (this is where the condition that the flow is subsonic is used), in such a way that the existence result is simply a consequence of the uniqueness result. Uniqueness is obtained by energy type boundary estimates combined with unique continuation arguments.

For the modified Helmholtz (24) problem the difficulty is that we do not know how to obtain the uniqueness result (this will be explained at the beginning of Section 3.4.3). That is why, to get around this obstacle, we shall use the analytic Fredholm theory, which requires to extend the problem to complex frequencies. This is for putting the problem in the adequate abstract framework, that we shall need the stronger admissibility condition (21) for the flow $v_0$. This is also because we use this theory that we have to exclude the set of possible resonances $R_{ex}$ for the existence and uniqueness result.

The rest of this section is organized as follows. Section 3.3 is devoted to a recap of the existent theory for the convected Helmholtz equation. The main section is Section 3.4 where we develop the proof of Theorem 3 via a perturbation approach explained above. Finally, in Section 3.5, we give a precise reinterpretation (when $d = 2$) of the admissibility condition as a small vorticity condition.
### 3.3. Resolution of the convected Helmholtz equation

As said above, the convected Helmholtz equation (24) is the one that must be solved when the mean flow is irrotational. However, from the mathematical point of view, this equation makes sense even for a non potential flow, $\omega_0 := \nabla \times v_0 \neq 0$, even though it is physically meaningful only when $\omega_0 = 0$.

#### 3.3.1. Weak formulation of the problem

It is through the variational formulation that the boundary value problem will acquire a precise sense. In (22), in comparison with the classical Helmholtz equation, the only difference is a second order term, namely $\rho_0 D_\omega (c_0^{-2} D_\omega \varphi)$, which replaces the usual zero order term $-\omega^2 c_0^{-2} \rho_0 \varphi (D_\omega \varphi = -i\omega\varphi$ if $v_0 = 0)$.

**Proposition 1.** If $\mathrm{div}(\rho_0 v_0) = 0$, the weak formulation of (22) reads, setting $V = H^1(\Omega)$,

$$ \text{Find } \varphi \in V \text{ such that } \quad \forall \psi \in V, \quad a(\omega; \varphi, \psi) = \ell(\psi), \qquad (25) $$

where the sesquilinear form $a(\omega; \varphi, \psi)$ and the antilinear form $\ell(\psi)$ are defined on $V$ by:

$$ \left\{ \begin{array}{l} a(\omega; \varphi, \psi) := \displaystyle\int_{\Omega} \rho_0 \left[ \nabla\varphi \cdot \nabla\bar{\psi} - (M_0 \cdot \nabla\varphi)(M_0 \cdot \nabla\bar{\psi}) \right] - \omega^2 \displaystyle\int_{\Omega} \rho_0 c_0^{-2} \varphi \bar{\psi} \\ \qquad + i\omega \displaystyle\int_{\Omega} \rho_0 c_0^{-1} \left[ \varphi(M_0 \cdot \nabla\bar{\psi}) - (M_0 \cdot \nabla\varphi)\bar{\psi} \right] - i\omega \displaystyle\int_{\partial\Omega} \lambda \rho_0 c_0^{-1} \varphi \bar{\psi}, \\ \\ \ell(\psi) := \displaystyle\int_{\Omega} \rho_0 f \bar{\psi}. \end{array} \right. \qquad (26) $$

**Proof.** It is quite standard, multiplying (22)(i) by $\bar{\psi}$, for $\psi \in H^1(\Omega)$ and integrating over $\Omega$ yields, after using Green's formula,

$$ \int_{\Omega} \rho_0 (\nabla\varphi - c_0^{-2} D_\omega \varphi v_0) \cdot \nabla\bar{\psi} - i\omega \int_{\Omega} \rho_0 c_0^{-2} D_\omega \varphi \bar{\psi} - \langle \rho_0 (\nabla\varphi - c_0^{-2} D_\omega \varphi v_0) \cdot n, \bar{\psi} \rangle_{\partial\Omega} = \int_{\Omega} \rho_0 f \bar{\psi}. $$

Then, to obtain (25), it suffices to first use (22)(ii) in order to get

$$ \langle \rho_0 (\nabla\varphi - c_0^{-2} D_\omega \varphi v_0) \cdot n, \bar{\psi} \rangle_{\partial\Omega} = i\omega \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} \varphi \bar{\psi}, $$

and then observe that, by definition of $D_\omega$ and $c_0^{-1} v_0 = M_0$,

$$ \left\{ \begin{array}{l} -c_0^{-2} D_\omega \varphi (v_0 \cdot \nabla \bar{\psi}) = i\omega c_0^{-1} \varphi (M_0 \cdot \nabla \bar{\psi}) - (M_0 \cdot \nabla \varphi)(M_0 \cdot \nabla \bar{\psi}), \\ \\ -i\omega c_0^{-2} D_\omega \varphi \bar{\psi} = -i\omega c_0^{-1} (M_0 \cdot \nabla \varphi) \bar{\psi} - \omega^2 c_0^{-2} \varphi \bar{\psi}, \end{array} \right. $$

which we substitute into (30).

<figure><img src="image_2.png" /></figure>

#### 3.3.2. Existence and uniqueness result

In order to get tighter inequalities by avoiding factors such as $\rho_+/\rho_-$, we choose to introduce the weighted norms $\| \cdot \|_{L^2(\Omega, \rho_0)}$ and $\| \cdot \|_{H^1(\Omega, \rho_0)}$, on $L^2(\Omega)$ and $H^1(\Omega)$, defined for $\phi \in L^2(\Omega)$,
$\psi \in H^1(\Omega)$ by

$$ \|\phi\|_{L^2(\Omega, \rho_0)} := \left( \int_{\Omega} \rho_0 |\phi|^2 \right)^{1/2}, \quad \|\psi\|_{H^1(\Omega, \rho_0)} := \left( \|\psi\|_{L^2(\Omega, \rho_0)}^2 + \|\nabla \psi\|_{L^2(\Omega, \rho_0)^d}^2 \right)^{1/2}. $$

The assumption (2) on $\rho_0$ ensures that these norms are equivalent to the usual one used on $L^2(\Omega)$ and $H^1(\Omega)$.

**Theorem 2.** The potential flow problem (22) is well-posed under the assumption (13) and the condition that the flow is strictly subsonic, namely

$$ \sup_{\mathbf{x} \in \Omega} |\mathbf{M}_0(\mathbf{x})|^2 < 1. \qquad (27) $$

**Proof.** It is carried out in two steps: (i) the problem is of Fredholm type and (ii) and admits at most one solution.

For (i), we decompose (artificially) the bilinear form $a(\omega; \varphi, \psi)$ as the sum of two terms

$$ a(\omega; \varphi, \psi) = a_*(\omega; \varphi, \psi) + c(\omega; \varphi, \psi), $$

where (note that we add and subtract artificially the term $\int_{\Omega} \rho_0 \varphi \overline{\psi}$):

$$ \left\{ \begin{aligned} a_*(\omega; \varphi, \psi) &:= \int_{\Omega} \rho_0 [\nabla\varphi \cdot \nabla\overline{\psi} - (M_0 \cdot \nabla\varphi)(M_0 \cdot \nabla\overline{\psi}) + \varphi\overline{\psi}] - i\omega \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} \varphi\overline{\psi}, \\ c(\omega; \varphi, \psi) &:= i\omega \int_{\Omega} \rho_0 c_0^{-1} [\varphi(M_0 \cdot \nabla\overline{\psi}) - (M_0 \cdot \nabla\varphi)\overline{\psi}] - \int_{\Omega} \rho_0 (1 + \omega^2 c_0^{-2}) \varphi\overline{\psi}. \end{aligned} \right. $$

Next, we observe that, thanks to (27), $a_*(\omega; \varphi, \psi)$ is coercive in $H^1(\Omega)$. Indeed since the volume integral in the expression of $a_*(\omega; \varphi, \varphi)$ is real while the boundary integral is purely imaginary, we have

$$ \begin{aligned} |a_*(\omega; \varphi, \varphi)| &\ge \int_{\Omega} \rho_0 (|\nabla\varphi|^2 - |(M_0 \cdot \nabla)\varphi|^2 + |\varphi|^2) \\ &\ge (1 - \sup_{\mathbf{x} \in \Omega} |M_0(\mathbf{x})|^2) \|\varphi\|_{H^1(\Omega, \rho_0)}^2. \end{aligned} \qquad (28) $$

Finally the bilinear form $c(\varphi, \psi)$ as the following compactness property

$$ (\varphi_n, \psi_n) \underset{n \to \infty}{\longrightarrow} (\varphi, \psi) \text{ weakly in } H^1(\Omega)^2 \implies c(\omega; \varphi_n, \psi_n) \underset{n \to \infty}{\longrightarrow} c(\omega; \varphi, \psi). $$

This is a consequence of the compact embedding of $H^1(\Omega)$ into $L^2(\Omega)$ and the fact that $c(\omega; \varphi, \psi)$ is a sum of integrals of quadratic quantities in which only first order derivatives of one of the two functions $\varphi$ or $\psi$ appear. As a consequence, the problem falls under the Fredholm alternative [40,48].

For the point (ii) (uniqueness), let us introduce $\varphi \in H^1(\Omega)$ solution of (22) with $f = 0$:

$$ \forall \psi \in H^1(\Omega), \quad a(\omega; \varphi, \psi) = 0. $$

Then, choosing $\psi = \varphi$ and taking the imaginary part of this equality leads to:

$$ \Im[a(\omega; \varphi, \varphi)] = 0 = \omega \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} |\varphi|^2. $$
Since $\lambda > 0$ on $\gamma \subset \partial\Omega$ (assumption (13)), this shows that $\varphi = 0$ on $\gamma$. From the boundary condition (15)(i), we deduce that $\nabla\varphi \cdot n = 0$ on $\gamma$. As $\Omega$ is connected, we conclude, with a classical unique continuation argument for elliptic equations [34–36], that $\varphi = 0$ in $\Omega$. $\square$

## 3.4. Proof of the main result

### 3.4.1. Preamble

Let us start with the weak formulation of the coupled problem (14,15) (the natural extension of (25) obtained for the convected Helmholtz equation). Let us introduce the Hilbert spaces

$$V := H^1(\Omega), \quad M := H^{-}(v_0, \Omega), \quad \text{and} \quad L := L^2(\Omega)^d.$$

**Proposition 2.** The weak formulation of (14,15) is: find $(\varphi, \xi) \in V \times M$ such that

$$\left\{
\begin{array}{ll}
a(\omega; \varphi, \psi) + d(\xi, \psi) = \ell(\psi), & \forall \psi \in V, \quad (i) \\
b(\varphi, \zeta) + t(\omega; \xi, \zeta) = 0, & \forall \zeta \in L, \quad (ii)
\end{array}
\right.
\qquad (29)$$

where the sesquilinear form $a(\omega; \varphi, \psi)$ and the antilinear form $\ell(\psi)$ are defined in (26), while the other sesquilinear forms are given by

$$\left\{
\begin{aligned}
t(\omega; \xi, \zeta) &:= \int_{\Omega} (D_{\omega}\xi \cdot \bar{\zeta} + (\xi \cdot \nabla)v_0 \cdot \bar{\zeta}), \\
d(\xi, \psi) &:= \int_{\Omega} \rho_0 \xi \cdot \overline{\nabla\psi} - \int_{\Gamma_+} \rho_0 c_0^{-1}(v_0 \cdot n)(\xi \cdot n) \bar{\psi}, \\
b(\varphi, \zeta) &:= \int_{\Omega} (\omega_0 \times \nabla\varphi) \cdot \bar{\zeta}.
\end{aligned}
\right.$$

**Proof.** As (29)(ii) simply results from multiplication of (14)(ii) by $\bar{\zeta}$ and integration over $\Omega$, only (29)(i) deserves some comments. This equation is obtained as in the proof of Proposition 1, by multiplying (14)(i) by $\bar{\psi}$, integrating over $\Omega$ and using Green's formula, which gives

$$\int_{\Omega} \rho_0 (\nabla \varphi + \xi - c_0^{-2} D_{\omega} \varphi v_0) \cdot \nabla \bar{\psi} - i\omega \int_{\Omega} \rho_0 c_0^{-2} D_{\omega} \varphi \bar{\psi} - \langle \rho_0 (\nabla \varphi + \xi - c_0^{-2} D_{\omega} \varphi v_0) \cdot n, \bar{\psi} \rangle_{\partial\Omega} = \int_{\Omega} \rho_0 f \bar{\psi}. \qquad (30)$$

One then concludes as in the proof of Proposition 1 using (15)(i). $\square$

For the sequel, it is useful to rewrite (29) in an abstract operator form. To this purpose, via Riesz theorem, we introduce the linear operators associated to the sesquilinear forms in (29)

$$\left\{
\begin{array}{lll}
A(\omega) \in \mathcal{L}(V) & \text{s.t.} & (A(\omega)\varphi, \psi)_V = a(\omega; \varphi, \psi), \quad \forall (\varphi, \psi) \in V \times V, \\
T(\omega) \in \mathcal{L}(M, L) & \text{s.t.} & (T(\omega)\xi, \zeta)_L = t(\omega; \xi, \zeta), \quad \forall (\xi, \zeta) \in M \times L, \\
D \in \mathcal{L}(M, V) & \text{s.t.} & (D\xi, \psi)_V = d(\xi, \psi), \quad \forall (\xi, \psi) \in M \times V, \\
B \in \mathcal{L}(V, L) & \text{s.t.} & (B\varphi, \zeta)_L = b(\varphi, \zeta), \quad \forall (\varphi, \zeta) \in V \times L.
\end{array}
\right.
\qquad (31)$$
Then, with $\hat{f} \in V$ such that $\ell(\varphi) = (\hat{f}, \psi)_V$ for all $\psi$ in $V$, (29) rewrites

$$ \begin{pmatrix} A(\omega) & D \\ \mathbf{B} & \mathbf{T}(\omega) \end{pmatrix} \begin{pmatrix} \varphi \\ \boldsymbol{\xi} \end{pmatrix} = \begin{pmatrix} \hat{f} \\ \mathbf{0} \end{pmatrix}. \qquad (32) $$

**Remark 7.** Note that the operator $\mathbf{T}(\omega)$ is nothing but the transport differential operator $\mathbf{T}_0(\omega)$, see (9), acting in the space $\mathbf{M}$ and that $\mathbf{B}\varphi = \omega_0 \times \nabla\varphi$, since their definition simply uses the $L^2$-inner product (or $\mathbf{L}$-inner product). The interpretation of $A(\omega)$ and $D$ is less direct since they are defined through the $H^1$-inner product.

For solving (32), the most natural idea, is to try to extend the Fredholm type approach followed in Section 3.3 for the convected Helmholtz equation. However, a first obstacle is that the uniqueness proof in Theorem 2 is not generalisable (at least we did not succeed) to the coupled problem (14, 15). Indeed, let $(\varphi, \boldsymbol{\xi})$ be a solution of (14, 15) for $f = 0$. Taking $\psi = \varphi$ in (29)(i) gives, after taking the imaginary part

$$ \omega \int_{\Gamma} \lambda \rho_0 c_0^{-1} |\varphi|^2 = \mathfrak{F}m \left\{ - \int_{\Omega} \rho_0 \boldsymbol{\xi} \cdot \nabla \bar{\varphi} + \int_{\Gamma_+} \rho_0 c_0^{-1} (\mathbf{v}_0 \cdot \mathbf{n})(\boldsymbol{\xi} \cdot \mathbf{n}) \bar{\varphi}, \right\}. $$

We could conclude if, for instance, we could deduce from the transport equation (14)(ii) that the right hand side term is non-positive. Unfortunately, we did not succeed (even in the simple case of a laminar flow in a wave guide) which leads us to doubt that this is true in general. This is why we have chosen to use analytic Fredholm theory as explained in Section 3.2.

The idea is to extend the Goldstein's problem (14, 15) to complex values of the frequency $\omega$, especially to

$$ \omega \in \mathbb{C}_{\beta}^{+} := \{z \in \mathbb{C} / \mathfrak{F}m(z) > -\beta\} $$

for $\beta > 0$ small enough: this will be made precise in the proof, see Section 3.4. As we shall see, the key properties of this domain is that it is connected, contains the real axis as well as the semi-imaginary one $\{i\omega_i / \omega_i > 0\}$.

This will permit us to apply the following abstract result from analytic Fredholm theory. For instance, from [48]-Theorem 8.92:

**Theorem 3 (analytic Fredholm).** Let $V$ a Hilbert space and $G \subset \mathbb{C}$ a domain (open and connected). Suppose that the map $\lambda \in G \mapsto B(\lambda) \in \mathcal{L}(V)$ is analytic and such that $\forall \lambda \in G$, $B(\lambda)$ is compact. Then, either,

(1) $(I - B(\lambda))^{-1}$ exists for no $\lambda \in G$,

(2) There exists $S \subset G$, discrete (i.e. with no limit point in $G$) such that for all $\lambda \in G \setminus S$, $(I - B(\lambda))^{-1}$ exists and the map $\lambda \in G \setminus S \mapsto (I - B(\lambda))^{-1}$, is analytic.

from which one easily deduces the following corollary that is more directly fitted to our settings:

**Corollary 1 (analytic Fredholm (invertible case)).** Let $V$ be a Hilbert space and $G \subset \mathbb{C}$ a domain. Let $A^g(\omega)$, $\omega \in G$, be a family of bounded linear operators in $V$ and assume that $A^g(\omega) = A_*^g(\omega) + C^g(\omega)$ with $A_*^g(\omega), C^g(\omega) \in \mathcal{L}(V)$, where
(1) $A_*^g : \omega \in G \mapsto A_*^g(\omega)$ and $C^g : \omega \in G \mapsto C^g(\omega)$ are analytic,

(2) $\forall \omega \in G$, $A_*^g(\omega)$ is invertible and $C^g(\omega)$ is compact,

(3) $\exists \omega_* \in G$ such that $A^g(\omega_*)$ is invertible.

Then, there exists a discrete set $S \subset G$ such that $\forall \omega \in G \setminus S$, $A^g(\omega)$ is invertible.

**Remark 8.** • Here, the analyticity of $A^g : \omega \in G \mapsto A^g(\omega) \in \mathcal{L}(V)$ has to be understood in the sense of the operator norm, i.e.

$$ \forall \omega \in G, \lim_{\substack{z \to 0, \\ z \in \mathbb{C}^*}} \frac{1}{z}(A^g(\omega + z) - A^g(\omega)) \text{ exists in } \mathcal{L}(V). $$

• A Fredholm operator is a bounded linear operator which kernel and a supplementary of its range are finite dimensional. Its index is the difference between these two dimensions. An invertible operator is a Fredholm operator of index 0, and because the Fredholm property and the index value are stable by adding a compact operator, assumption (2) of the Corollary 1 implies that $A^g(\omega)$ is a Fredholm operator of index 0.

The above result can not be directly applied to coupled problem (32) in $(\varphi, \xi)$. We shall apply it to the problem in $\varphi$ alone obtained formally by eliminating $\xi$. In other words, we perform a Schur complement

$$ \xi = \xi(\varphi) = -\mathbf{T}(\omega)^{-1}\mathbf{B}\varphi, \qquad (33) $$

so that the equation governing $\varphi$ is given by the “modified” convected Helmholtz equation:

$$ A_r(\omega)\varphi = f, \quad \text{where } A_r(\omega) := A(\omega) - D\mathbf{T}(\omega)^{-1}\mathbf{B}. $$

and we aim at apply Corollary 1 to $V = H^1(\Omega)$ and $A^g(\omega) = A_r(\omega)$.

**Remark 9.** Note that, going back to the notation (23) of Section 3.2,

$$ -\mathbf{T}(\omega)^{-1}\mathbf{B}\varphi = \mathbf{S}_0(\omega, \nabla\varphi). $$

In Section 3.4.2, we first show that the elimination of $\xi$ is possible, namely that (33) has a sense (this is of course related to the invertibility of the transport equation). In Section 3.4.3, we show that the family $A_r(\omega)$ satisfies the assumptions (1) and (2) of Corollary 1. Finally, in Section 3.4.4, we conclude the proof by showing that the assumption (3) is also satisfied.

### 3.4.2. Elimination of $\xi$ and modified convected Helmholtz equation

As this section is dedicated to the elimination of $\xi$ from the transport equation (7)(i), let us begin with the Theorem 4 which states the invertibility of the transport operator, introduced in (9), acting on $M$, i.e. $T_0(\omega) : M \subset L \to L$. Whereas the issue of the invertibility of this operator has been previously treated in [8], it has to be adapted here to complex frequencies $\omega$. For completeness, let us recall the definition of a $\Omega$-filling flow $v_0$ which is the key assumption of the Theorem 4.

The characteristic field $\chi : \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}^d$ associated to $v_0 \in C^1(\mathbb{R}^d)^d$ is defined from the solutions of the following family of differential equations:

$$ \forall (t, \mathbf{b}) \in \mathbb{R} \times \mathbb{R}^d, \quad \left\{ \begin{array}{l} \displaystyle \frac{\partial \chi}{\partial t}(t, \mathbf{b}) = v_0(\chi(t, \mathbf{b})), \\ \\ \chi(0, \mathbf{b}) = \mathbf{b}, \end{array} \right. $$
(we do not use the term “flow” from differential equation theory to avoid confusion with the flow $v_0$). Using $\chi$, we can now precise the Definition 1 of a $\Omega$-filling flow $v_0$:

**Definition 3** ($\Omega$-filling flow and life time in $\Omega$). The flow $v_0$ is said $\Omega$-filling if there exists a time $T^* \in \mathbb{R}^+$ such that for all $x \in \Omega$:

$$ \exists(t, \mathbf{b}) \in [0, T^*] \times \Gamma_- / \chi(t, \mathbf{b}) = \mathbf{x} \quad \text{and} \quad \forall \tau \in ]0, t], \quad \chi(\tau, \mathbf{b}) \in \Omega. \quad (34) $$

For such a flow, its life time in $\Omega$, denoted $t^*(v_0, \Omega)$, is the infimum of all constants $T^*$ keeping (34) valid.

We then have the following theorem:

**Theorem 4** (Invertibility of $\mathcal{T}_0(\omega)$ for $\omega \in \mathbb{C}$). If the flow $v_0$ is $\Omega$-filling and satisfies $\text{div}(\rho_0 v_0) = 0$, with the life time $t^*(v_0, \Omega)$ in $\Omega$, then, for all $\omega \in \mathbb{C}$, the time harmonic transport operator $\mathcal{T}_0(\omega) : M \subset L \to L$ defined by (9) is invertible and the following estimates hold: denoting $\omega_i = \Im m \omega$, for all $g \in L$,

$$ \begin{aligned} \| \mathcal{T}_0^{-1}(\omega) g \|_{L^2(\Omega, \rho_0)d} &\le C_0(v_0, \Omega, \omega_i) \|g\|_{L^2(\Omega, \rho_0)d} \quad (i), \\ \| \mathcal{T}_0^{-1}(\omega) g \|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))} &\le C_+(v_0, \Omega, \omega_i) \|g\|_{L^2(\Omega, \rho_0)d} \quad (ii), \end{aligned} \quad (35) $$

where the weighted $L^2$-norm along $\Gamma_+$ is given by, remembering that $v_0 \cdot n > 0$ on $\Gamma_+$,

$$ \|w\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}^2 := \int_{\Gamma_+} |w|^2 \rho_0(v_0 \cdot n) d\sigma, $$

and $C_0(v_0, \Omega, \omega_i)$ and $C_1(v_0, \Omega, \omega_i)$ are respectively given, with $(\Phi, \Psi)$ defined in (20), by

$$ \left\{ \begin{aligned} C_0(v_0, \Omega, \omega_i) &:= t^*(v_0, \Omega) \Phi(2 t^*(v_0, \Omega) (\|\nabla v_0\|_{L^\infty} - \omega_i)), \quad (i) \\ C_+(v_0, \Omega, \omega_i) &:= \Psi(2 t^*(v_0, \Omega) (\|\nabla v_0\|_{L^\infty} - \omega_i)). \quad (ii) \end{aligned} \right. \quad (36) $$

**Remark 10.** As the map $\Phi: \mathbb{R} \to \mathbb{R}$ is increasing (Lemma 1), (36)(i) implies that the function $\omega_i \in \mathbb{R} \mapsto C_0(v_0, \Omega, \omega_i)$ is (strictly) decreasing. In particular

$$ \forall \omega_i \ge 0, \quad C_0(v_0, \Omega, \omega_i) \le C_0(v_0, \Omega, 0) = t^*(v_0, \Omega) \Phi(2 t^*(v_0, \Omega) \|\nabla v_0\|_{L^\infty}). $$

Moreover, as $\Phi(s)$ and $\Psi(s)$ tend to 0 when $s \to -\infty$ (Lemma 1 again), it is also clear from formulas (36) that

$$ C_0(v_0, \Omega, \omega_i) \to 0 \quad \text{and,} \quad C_+(v_0, \Omega, \omega_i) \to 0 \quad \text{when} \quad \omega_i \to +\infty. $$

**Proof.** The proof of Theorem 4 is a straightforward adaptation of the same result for real frequencies detailed in [8]. The key point consists in obtaining the following two a priori estimates (obviously related to (35))

$$ \begin{aligned} \forall \xi \in M, \quad & \|\xi\|_{L^2(\Omega, \rho_0)d} \le C_0(v_0, \Omega, \omega_i) \|\mathcal{T}_0(\omega)\zeta\|_{L^2(\Omega, \rho_0)d}, \quad (i) \\ \forall \xi \in M, \quad & \|\xi\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))} \le C_+(v_0, \Omega, \omega_i) \|\mathcal{T}_0(\omega)\xi\|_{L^2(\Omega, \rho_0)d}. \quad (ii) \end{aligned} \quad (37) $$
Once the estimates (37) are established, the proof of the invertibility of $\mathcal{T}_0(\omega)$ essentially relies on functional analytic arguments and (37) leads to (35). We divide the proof into three steps.

**Step 1:** Preliminary material. We use the same technique of "characteristic" change of variables as in [8]. Thanks to the $\Omega$-filling property of $v_0$ and the fact that the boundary $\Gamma_-$ is piecewise $\mathcal{C}^1$, the function $\chi$ induces a piecewise-$\mathcal{C}^1$-diffeomorphism $\chi : \Theta \to \Omega$ on the fiber bundle

$$ \Theta := \{(t, \mathbf{b}) \in \mathbb{R}_+ \times \Gamma_- : \forall \tau \in (0, t], \chi(\tau, \mathbf{b}) \in \Omega\}. $$

In what follows, we endow $\Theta$ with the push-forward measure $\mu = (\rho_0 d\mathbf{x}) \circ \chi$ such that for any Lebesgue measurable set $B \subset \Theta$,

$$ \mu(B) := (\rho_0 d\mathbf{x}) [\chi(B)]. $$

This leads to the change of variables formula: given $f : \Omega \to \mathbb{C}$,

$$ \int_{\Theta} f \circ \chi(t, \mathbf{b}) \, d\mu(t, \mathbf{b}) = \int_{\Omega} f(\mathbf{x}) \rho_0(\mathbf{x}) d\mathbf{x}. \quad (38) $$

A classical consequence of $\mathrm{div}(\rho_0 v_0) = 0$, [4, 16], is that the measure $d\mu(t, \mathbf{b})$ can be written as the product measure of $dt$ and the surfacic measure $d\sigma$ on $\Gamma_-$ (which means nothing but the fact that the volume is preserved along the streamlines):

$$ d\mu(t, \mathbf{b}) = dt d\sigma(\mathbf{b}), $$

and thus the integration over $\Theta$ reads, given $g : \Theta \to \mathbb{C}$,

$$ \int_{\Theta} g(t, \mathbf{b}) \, d\mu(t, \mathbf{b}) = \int_{\Theta} g(t, \mathbf{b}) \, dt \, d\sigma(\mathbf{b}) = \int_{\Gamma_{-}} \left( \int_{0}^{t(\mathbf{b})} g(t, \mathbf{b}) \, dt \right) d\sigma(\mathbf{b}), \quad (39) $$

where $t(\mathbf{b}) := \sup\{t \ge 0 : \forall \tau \in (0, t], \chi(t, \mathbf{b}) \in \Omega\}$ is by definition the life time in $\Omega$ of $\mathbf{b} \in \Gamma_-$. Note that this notion is related to the global lifetime in $\Omega$ via

$$ t^*(v_0, \Omega) = \sup_{\mathbf{b} \in \Gamma_{-}} t(\mathbf{b}). \quad (40) $$

As a consequence of formula (38), the operator $\mathcal{S} \in \mathcal{L}(L^2(\Omega, \rho_0)^d, L^2(\Theta, \mu)^d)$ defined by

$$ \forall \boldsymbol{\xi} \in L^2(\Omega)^d, \quad \forall (t, \mathbf{b}) \in \Theta, \quad \mathcal{S}\boldsymbol{\xi}(t, \mathbf{b}) := \boldsymbol{\xi}(\chi(t, \mathbf{b})) $$

is an isometry, namely:

$$ \forall \boldsymbol{\xi} \in L^2(\Omega, \rho_0)^d, \quad \|\mathcal{S}\boldsymbol{\xi}\|_{L^2(\Theta)^d} = \|\boldsymbol{\xi}\|_{L^2(\Omega, \rho_0)^d}. \quad (41) $$

The main interest of the change of variables $\mathbf{x} = \chi(t, \mathbf{b})$ is that it simplifies the expression of the transport operator in $\Theta$ variables. More precisely, one observes that setting

$$ \widehat{M} := \{ \widehat{\zeta} \in L^2(\Theta, \mu)^d / \frac{\partial \widehat{\zeta}}{\partial t} \in L^2(\Theta, \mu)^d, \forall \mathbf{b} \in \Gamma_{-}, \widehat{\zeta}(0, \mathbf{b}) = \mathbf{0} \}, $$

then $\boldsymbol{\xi} \in \mathcal{M}$ if and only if $\widehat{\boldsymbol{\xi}} := \mathcal{S}\boldsymbol{\xi} \in \widehat{M}$. Moreover, one has the formula $(v_0 \cdot \nabla)$ simply becomes the $t$-derivative, this is pure computation, see [8] for $d = 2$)

$$ \forall \boldsymbol{\zeta} \in \mathcal{M}, \quad \mathcal{S}((v_0 \cdot \nabla)\boldsymbol{\zeta}) = \frac{\partial}{\partial t}(\mathcal{S}\boldsymbol{\zeta}). \quad (42) $$
As a consequence if we introduce the operator $\mathcal{Q}_0(\omega)$ defined by

$$ \forall \hat{\zeta} \in \hat{M}, \quad \mathcal{Q}_0(\omega) \hat{\zeta} := \frac{\partial \hat{\zeta}}{\partial t} - i\omega \hat{\zeta} + \hat{J}_0 \hat{\zeta}, $$

where $\hat{J}_0(t, \mathbf{b}) \in \mathcal{L}(\mathbb{R}^b)$ is defined as

$$ \hat{J}_0(t, \mathbf{b}) = J_0(\chi(t, \mathbf{b})), \quad \text{where } J_0(\mathbf{x}) = D\mathbf{v}_0(\mathbf{x}), \text{ i.e. } (J_0)_{ij}(\mathbf{x}) = \frac{\partial \mathbf{v}_{0,j}}{\partial x_i}(\mathbf{x}), $$

one has the commutation property (this is pure computation, see [8] for $d=2$)

$$ \forall \xi \in \mathcal{M}, \quad \mathcal{S} \mathcal{T}_0(\omega) \xi = \mathcal{Q}_0(\omega) \mathcal{S} \xi. \qquad (43) $$

**Step 2.** Proof of the volume estimate (37)(i).

From (43) and the isometry result (41), it is clear that proving (37)(i) amounts to proving

$$ \forall \hat{\xi} \in \hat{M}, \quad \|\hat{\xi}\|_{L^2(\Theta, \mu)^d} \le C_0(\mathbf{v}_0, \Omega, \omega) \|\mathcal{Q}_0(\omega)\hat{\xi}\|_{L^2(\Theta, \mu)^d} \qquad (44) $$

Of course, by a density-continuity argument, it suffices to establish (44) when

$$ \hat{\xi} \in D_-(\Theta) := \{ \zeta \in \hat{M} / \text{ a.e. } \mathbf{b} \in \Gamma_-, t \to \zeta(t, \mathbf{b}) \in \mathcal{C}^1([0, t(\mathbf{b}))] \}. $$

which is a dense subset of $\hat{M}$. For such a $\hat{\xi}$, we write

$$ \begin{aligned} \hat{\xi}(t, \mathbf{b}) e^{-i\omega t} &= \int_0^t \frac{\partial}{\partial \tau} (\hat{\xi}(\tau, \mathbf{b}) e^{-i\omega\tau}) d\tau = \int_0^t e^{-i\omega\tau} \left( \frac{\partial \hat{\xi}}{\partial \tau}(\tau, \mathbf{b}) - i\omega \hat{\xi}(\tau, \mathbf{b}) \right) d\tau \\ &= \int_0^t e^{-i\omega\tau} \mathcal{Q}_0(\omega) \hat{\xi}(\tau, \mathbf{b}) d\tau + \int_0^t e^{-i\omega\tau} \hat{J}_0(\tau, \mathbf{b}) \hat{\xi}(\tau, \mathbf{b}) d\tau. \end{aligned} $$

Then, using $|e^{-i\omega t} \hat{\xi}(t, \mathbf{b})| = e^{\omega_i t} |\hat{\xi}(t, \mathbf{b})|$ we get

$$ e^{\omega_i t} |\hat{\xi}(t, \mathbf{b})| \le \int_0^t e^{\omega_i \tau} |\mathcal{Q}_0(\omega) \hat{\xi}(\tau, \mathbf{b})| d\tau + \int_0^t e^{\omega_i \tau} |\hat{\xi}(\tau, \mathbf{b})| |\tilde{J}_0(\tau, \mathbf{b})| d\tau. $$

Gronwall's lemma then provides that

$$ e^{\omega_i t} |\hat{\xi}(t, \mathbf{b})| \le \int_0^t \exp \left( \int_\tau^t |\tilde{J}_0(\tau', \mathbf{b})| d\tau' \right) e^{\omega_i \tau} |\mathcal{Q}_0(\omega) \hat{\xi}(\tau, \mathbf{b})| d\tau, $$

and Cauchy-Schwartz inequality leads to

$$ |\hat{\xi}(t, \mathbf{b})|^2 \le \mathcal{I}(t, \mathbf{b}) \int_0^t |\mathcal{Q}_0 \hat{\xi}(\tau, \mathbf{b})|^2 d\tau, \qquad (45) $$

where $\mathcal{I}(\omega_i; t, \mathbf{b}) := \int_0^t e^{2 \int_\tau^t |\tilde{J}_0(\tau', \mathbf{b})| d\tau'} e^{-2\omega_i(t-\tau)} d\tau$. Since $|\tilde{J}_0(\tau', \mathbf{b})| \le \|\nabla \mathbf{v}_0\|_\infty$, we get

$$ \mathcal{I}(\omega_i; t, \mathbf{b}) \le \mathcal{I}_*(\omega_i; t) := \int_0^t e^{2(\|\nabla \mathbf{v}_0\|_\infty - \omega_i)(t-\tau)} d\tau = t \Psi(2t(\|\nabla \mathbf{v}_0\|_\infty - \omega_i))^2, \qquad (46) $$
where the last equality results from an explicit computation and definition of $\Psi$. Finally, using (46) in (45) and integrating the resulting inequality over $\Theta$ gives

$$
\begin{aligned}
\|\hat{\boldsymbol{\xi}}\|_{L^2(\Theta, \mu)^d}^2 &= \int_{\Gamma_-} \int_0^{t(b)} |\hat{\boldsymbol{\xi}}(t, \boldsymbol{b})|^2 dt d\sigma(\boldsymbol{b}) \\
&\le \int_{\Gamma_-} \int_0^{t(b)} \left( \mathcal{I}_*(\omega_i; t) \int_0^t |\boldsymbol{Q}_0 \hat{\boldsymbol{\xi}}(\tau, \boldsymbol{b})|^2 d\tau \right) dt d\sigma(\boldsymbol{b}) \\
&\le \int_{\Gamma_-} \left( \int_0^{t(b)} |\boldsymbol{Q}_0 \hat{\boldsymbol{\xi}}(\tau, \boldsymbol{b})|^2 d\tau \right) \left( \int_0^{t(b)} \mathcal{I}_*(\omega_i; t) dt \right) d\sigma(\boldsymbol{b}) \\
&\le \left( \int_0^{t^*(v_0, \Omega)} \mathcal{I}_*(\omega_i, t) dt \right) \left( \int_{\Gamma_-} \int_0^{t(b)} |\boldsymbol{Q}_0 \hat{\boldsymbol{\xi}}(\tau, \boldsymbol{b})|^2 d\tau d\sigma(\boldsymbol{b}) \right) \quad \text{(by (40))} \\
&\le \left( \int_0^{t^*(v_0, \Omega)} \mathcal{I}_*(\omega_i, t) dt \right) \|\boldsymbol{Q}_0(\omega) \hat{\boldsymbol{\xi}}\|_{L^2(\Theta, \mu)^d}^2,
\end{aligned}
$$

and an exact computation with the expression (46) of $\mathcal{I}^*(t)$ gives

$$ \int_{0}^{t^*(v_0, \Omega)} I_*(\omega_i, t) dt = C_0(v_0, \Omega, \omega_i)^2, $$

which leads to the desired estimate (44).

**Step 3.** Proof of the boundary estimate (37)(ii).

We first observe that, by Green's formula,

$$ \|\boldsymbol{\xi}\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}^2 := \int_{\Gamma^+} \rho_0 |\boldsymbol{\xi}|^2 (v_0 \cdot n) d\sigma = 2 \Re \int_{\Omega} \rho_0 (v_0 \cdot \nabla) \boldsymbol{\xi} \cdot \bar{\boldsymbol{\xi}} $$

Thus using the change of variables formula (38), (39) and (42)

$$ \|\boldsymbol{\xi}\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}^2 = 2 \Re \int_{\Theta} \frac{\partial \hat{\boldsymbol{\xi}}}{\partial t} \cdot \bar{\hat{\boldsymbol{\xi}}} d\mu(t, \boldsymbol{b}) = 2 \Re \int_{\Gamma_-} \int_0^{t(b)} \frac{\partial \hat{\boldsymbol{\xi}}}{\partial t} \cdot \bar{\hat{\boldsymbol{\xi}}} dt d\sigma(\boldsymbol{b}). $$

Thus, using again (45) and the inequality $\mathcal{I}(\omega_i; t(\boldsymbol{b}), \boldsymbol{b}) \le \mathcal{I}^*(\omega_i, t^*(\boldsymbol{v}_0, \Omega)) \le \mathcal{I}^*(\omega_i, t^*(\boldsymbol{v}_0, \Omega))$,

$$
\begin{aligned}
\|\boldsymbol{\xi}\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}^2 &= \int_{\Gamma_-} |\hat{\boldsymbol{\xi}}(t(\boldsymbol{b}), \boldsymbol{b})|^2 d\sigma(\boldsymbol{b}) \\
&\le \int_{\Gamma_-} \mathcal{I}(t(\boldsymbol{b}), \boldsymbol{b}) \left( \int_0^{t(b)} |\boldsymbol{Q}_0 \hat{\boldsymbol{\xi}}(\tau, \boldsymbol{b})|^2 d\tau \right) d\sigma(\boldsymbol{b}) \\
&\le \mathcal{I}_*(\omega_i, t^*(v_0, \Omega)) \int_{\Gamma_-} \int_0^{t(b)} |\boldsymbol{Q}_0 \hat{\boldsymbol{\xi}}(\tau, \boldsymbol{b})|^2 d\tau d\sigma(\boldsymbol{b}) \\
&= C_+(v_0, \Omega, \omega_i)^2 \|\boldsymbol{T}_0(\omega) \hat{\boldsymbol{\xi}}\|_{L^2(\Omega, \rho_0)^d},
\end{aligned}
$$
by definition of $C_+(v_0, \Omega, \omega_i)^2$.

<figure><img src="image_3.png" /></figure>

We can now apply Theorem 4 to reduce the Goldstein problem (14, 15) to the “modified” convected Helmholtz problem (24) with unknown $\varphi \in V$ by giving a rigorous sense to

$$ \mathbf{S}_0(\omega; \nabla \varphi) := \mathcal{T}_0^{-1}(\omega_0 \times \nabla \varphi) \in \mathcal{H}, \quad \forall \varphi \in V, \qquad (47) $$

since $\omega_0 \times \nabla \varphi \in \mathcal{L}$. We deduce in particular from the estimates, the continuity estimates

$$ \forall \varphi \in V, \quad \|\mathbf{S}_0(\omega; \nabla \varphi)\|_{L^2(\Omega, \rho_0)^d} \le C_0(v_0, \Omega, \omega_i) \|\omega_0\|_{L^\infty} \|\nabla \varphi\|_{L^2(\Omega, \rho_0)^d}, \quad (i) \qquad (48) $$

$$ \forall \varphi \in V, \quad \|\mathbf{S}_0(\omega; \nabla \varphi)\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))} \le C_+(v_0, \Omega, \omega_i) \|\omega_0\|_{L^\infty} \|\nabla \varphi\|_{L^2(\Omega, \rho_0)^d}, \quad (ii) $$

The next proposition gives the weak formulation of the reduced Goldstein's problem (24).

**Proposition 3.** If the flow $v_0$ is admissible and if $\mathrm{div}(\rho_0 v_0) = 0$, the weak formulation of (24) reads, with $\ell(\psi)$ defined as in (26):

$$ \text{For } f \in L^2(\Omega), \text{ find } \varphi \in V \text{ s.t. } \forall \psi \in V, \quad a^g(\omega; \varphi, \psi) = \ell(\psi), \qquad (49) $$

where the sesquilinear form $a^g(\omega, \varphi, \psi)$ is defined on $V$ by:

$$ a^g(\omega; \varphi, \psi) := a(\omega; \varphi, \psi) + d(\mathbf{S}_0(\omega; \nabla \varphi), \psi) \qquad (50) $$

where $a(\omega; \varphi, \psi)$ and $d(\omega; \varphi, \psi)$ are the sesquilinear forms defined in (26) and (31).

**Proof.** It is a straightforward consequence of weak formulation (14, 15) of the Goldstein's problem. Indeed, (15) means, via Theorem (4), that $\xi = \mathbf{S}_0(\omega; \nabla \varphi) \in \mathcal{H}$ that we substitute into (14) to get (49). $\square$

<figure><img src="image_4.png" /></figure>

### 3.4.3. Fredholm analytic property

This section is dedicated to show that the assumptions (1) and (2) of the Corollary 1 are satisfied for the operator $A^g(\omega) := A_r(\omega)$ defined in the preamble 3.4.1. The check of the assumption (1) is easy as the sesquilinear forms involved depend only polynomially on $\omega$. We have the following proposition:

**Proposition 4 (analyticity).** The map $\omega \in \mathbb{C} \mapsto A^g(\omega) \in \mathcal{L}(V)$ is analytic.

**Proof.** Using the notations of the preamble, as the sesquilinear forms $a$ and $t$ defined respectively on $V \times V$ and $H \times L$ have a polynomial dependence in the variable $\omega \in \mathbb{C}$, the associated operators $A(\omega)$ and $T(\omega)$ depends also polynomially, thus analytically, on $\omega$. Then, by standard results, if $\omega \mapsto T(\omega)$ is analytic and $T(\omega)$ is invertible for all $\omega$, the maps $\omega \mapsto T(\omega)^{-1}$ is also analytic. We finally deduce the analyticity of the mapping $\omega \mapsto A^g(\omega) := A(\omega) - D T(\omega)^{-1} B$ as a sum and composition of analytic maps. $\square$

<figure><img src="image_5.png" /></figure>

We now investigate the Fredholmness of $A^g(\omega)$ to verify assumption (2). In the proof of Theorem 2, we proved the Fredholmness of the convected Helmholtz problem by artificially decomposing its sesquilinear form $a$ as $a = a_* + c$ and showed that $a_*$ is coercive and $c$ is associated with a compact
operator. To use a similar approach for $a^g$, it is first worthwhile mentioning that the first part of
the bilinear form $d(S_0(\omega; \nabla\varphi), \psi)$ namely

$$
\int_{\Omega} \rho_0 \mathbf{S}_0(\omega; \nabla \varphi) \cdot \nabla \bar{\psi}, \qquad (51)
$$

is not compact in $V = H^1(\Omega)$. This is due to the fact that the operator $T_0$ is regularizing along the streamlines (this is estimate (35)) but not along the transverse directions. As a consequence, (51) needs to be incorporated in the "coercive part" of the decomposition of $a^g$. This leads us to the following decomposition

$$
\forall (\varphi, \psi) \in V \times V, \quad a^g(\omega; \varphi, \psi) = a_*^g(\omega; \varphi, \psi) + c^g(\omega; \varphi, \psi),
$$

$$
\text{where} \quad \left\{
\begin{aligned}
a_{*}^{g}(\omega; \varphi, \psi) &:= a_{*}(\omega; \varphi, \psi) + \int_{\Omega} \rho_{0} \mathbf{S}_{0}(\omega; \nabla \varphi) \cdot \nabla \bar{\psi}, \\
c^{g}(\omega; \varphi, \psi) &:= c(\omega; \varphi, \psi) - \int_{\Gamma_{+}} \rho_{0}(\mathbf{M}_{0} \cdot \mathbf{n}) \mathbf{S}_{0}(\omega; \nabla \varphi) \cdot \mathbf{n} \bar{\psi}.
\end{aligned}
\right.
$$

Let $A_*(\omega), C(\omega), A_*^g(\omega), C^g(\omega)$ in $\mathcal{L}(V)$ be the operators defined, via Riesz Theorem, by

$$
\forall \varphi, \psi \in V, \quad \left\{
\begin{aligned}
a_*(\omega; \varphi, \psi) &= \langle A_*(\omega)\varphi, \psi \rangle_V, & a_*^g(\omega; \varphi, \psi) &= \langle A_*^g(\omega)\varphi, \psi \rangle_V, \\
c(\omega; \varphi, \psi) &= \langle C(\omega)\varphi, \psi \rangle_V, & c^g(\omega; \varphi, \psi) &= \langle C^g(\omega)\varphi, \psi \rangle_V.
\end{aligned}
\right.
$$

**Proposition 5** (Fredholmness). If the flow $v_0$ is admissible (Definition 2, which includes in particular (2)), there exists $\beta > 0$ such that for all $\omega \in C_\beta^+$:

(i) $A_{*}^{g}(\omega)$ is invertible,

(ii) $C^g(\omega)$ is a compact operator,

so that $A^g(\omega)$ is a Fredholm operator of index 0.

Proof.

(i) We have proved that $a_*(\omega; \cdot, \cdot)$ is coercive for real frequency $\omega$, see (28). If $\omega := \omega_r + i\omega_i$ ($\omega_r, \omega_i \in \mathbb{R}$) is complex, we have, for $\varphi \in H^1(\Omega)$,

$$
|a_*(\omega, \varphi, \varphi)| \ge \Re e(a_*(\omega, \varphi, \varphi)) = \int_{\Omega} \rho_0 (|\nabla\varphi|^2 - |M_0 \cdot \nabla\varphi|^2 + |\varphi|^2) + \omega_i \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} |\varphi|^2.
$$

As the additional term $a_*^g(\omega; \varphi, \varphi) - a_*(\omega, \varphi, \varphi)$ has a priori no sign, we simply bound it from below by minus its absolute value. Using Cauchy-Schwarz inequality, we get

$$
\begin{align*}
|a_*^g(\omega; \varphi, \varphi)| &= |a_*(\omega; \varphi, \varphi) + \int_{\Omega} \rho_0 \mathbf{S}_0(\omega; \nabla\varphi) \cdot \nabla\bar{\varphi}| \\
&\geq |a_*(\omega; \varphi, \varphi)| - \|\mathbf{S}_0(\omega; \nabla\varphi)\|_{L^2(\Omega, \rho_0)^d} \|\nabla\varphi\|_{L^2(\Omega, \rho_0)^d},
\end{align*}
$$

and thus,

$$
|a_{*}^{g}(\omega; \varphi, \psi)| \geq g(\omega_i) \|\psi\|_{H^1(\Omega, \rho_0)}^2 + \omega_i \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} |\psi|^2,
$$
where we have set

$$g(\omega_i) := 1 - \|\mathbf{M}_0\|_{L^\infty}^2 - \|\boldsymbol{\omega}_0\|_{L^\infty} C_0(\mathbf{v}_0, \Omega, \omega_i). \quad (52)$$

As $\omega_i \mapsto C_0(\mathbf{v}_0, \Omega, \omega_i)$ is decreasing (Remark 10), $\omega_i \mapsto g(\omega_i)$ is increasing and we can bound by below as follow:

$$|a_*^g(\omega; \varphi, \varphi)| \geq g(-\beta) \|\varphi\|_{H^1(\Omega, \rho_0)}^2 - \beta \int_{\partial\Omega} \lambda \rho_0 c_0^{-1} |\varphi|^2.$$

By boundedness of $\lambda$ and $c_0^{-1}$ and trace theorem, there exists $C > 0$ such that

$$\int_{\partial\Omega} \lambda \rho_0 c_0^{-1} |\varphi|^2 \leq C \|\varphi\|_{H^1(\Omega, \rho_0)}^2.$$

It follows that

$$\forall \omega \in \mathbb{C}_{\beta}^{+}, \quad |a_{*}^{g}(\omega; \varphi, \varphi)| \geq [g(-\beta) - C\beta] \|\varphi\|_{H^{1}(\Omega, \rho_{0})}^{2}. \quad (53)$$

By a continuity argument (36), we see that

$$\lim_{\beta \to 0} [g(-\beta) - C\beta] = g(0),$$

where, by definition of $g(\omega_i)$, (52), and $C_0(\mathbf{v}_0, \Omega, \omega_i)$, (36),

$$g(0) = 1 - \|\mathbf{M}_0\|_{L^\infty}^2 - \|\boldsymbol{\omega}_0\|_{L^\infty} t^*(\mathbf{v}_0, \Omega) \Phi(2t^*(\mathbf{v}_0, \Omega) \|\nabla \mathbf{v}_0\|_{L^\infty}) > 0,$$

where the strict inequality is nothing but (21). By continuity, we can find $\beta > 0$ (small enough) such that $g(-\beta) - C\beta > 0$. Thus, for such a $\beta$, (53) provides the coercivity of $a_*^g(\omega; \cdot, \cdot)$ for any $\omega \in \mathbb{C}_\beta^+$, uniformly with respect to $\omega \in \mathbb{C}_\beta^+$.

As a consequence, $A_*^g(\omega)$ is invertible for any $\omega \in \mathbb{C}_\beta^+$.

(ii) The compactness of the operator $C(\omega)$ associated to the sesquilinear form $c(\omega, \cdot, \cdot)$ has already been proved in the proof of Theorem 2. It remains to show the compacity of the operator $C^d(\omega) = C^g(\omega) - C(\omega)$ associated with the sesquilinear form

$$\varphi, \psi \in V, \quad c^d(\omega, \varphi, \psi) := \int_{\Gamma_+} \rho_0 (\mathbf{M}_0 \cdot \mathbf{n}) (\mathbf{S}_0(\omega; \nabla \varphi) \cdot \mathbf{n}) \bar{\psi}.$$

This will rely on a sharp continuity estimate for $c^d(\omega, \varphi, \psi)$ which will provide more than the simple continuity of $C^d(\omega)$.

First, by Cauchy-Schwarz inequality, $\mathbf{M}_0 = c_0^{-1} \mathbf{v}_0$ and boundedness of various coefficients, we have, for some $C > 0$,

$$|c^d(\omega, \varphi, \psi)| \leq \|c_0^{-1}\| \|(\mathbf{S}_0(\omega; \nabla \varphi) \cdot \mathbf{n})\|_{L^2(\Gamma_+, \rho_0(\mathbf{v}_0 \cdot \mathbf{n}))} \|\psi\|_{L^2(\Gamma_+, \rho_0(\mathbf{v}_0 \cdot \mathbf{n}))}$$

Therefore, using the trace inequality (48)(ii), we have, with $\omega_i := \Im m \omega_i$,

$$|c^d(\omega, \varphi, \psi)| \leq C_+(\mathbf{v}_0, \Omega, \omega_i) \|\boldsymbol{\omega}_0\|_{L^\infty} \|\nabla \varphi\|_{L^2(\Omega, \rho_0)^d} \|\psi\|_{L^2(\Gamma_+, \rho_0(\mathbf{v}_0 \cdot \mathbf{n}))}$$

and the compactness of $C^d(\omega)$ is consequence of the compactness of the map $\psi \in V \mapsto \psi|_{\Gamma_+} \in L^2(\Gamma_+)$ (or the compact embedding of $H^{\frac{1}{2}}(\Gamma_+)$ in $L^2(\Gamma_+)$).

□
### 3.4.4. *End of the proof: invertibility at one frequency* $\omega_*$

To fulfill the conditions of the Corollary 1, it remains to verify that the operator $A^g(\omega)$, is invertible at one frequency $\omega_*$ in the domain $C_\beta^*$ where the problem has been shown to be of Fredholm type in the previous section. This value will be found on the positive imaginary semi-axis provided that the imaginary part of $\omega_*$ is large enough.

**Remark 11.** The idea of looking at frequencies with a positive imaginary par $\omega + i\varepsilon$, $\varepsilon > 0$ is classical in particular for the standard Helmholtz equation: it corresponds to the limiting absorption procedure. Looking at $\omega + i\varepsilon$ can be interpreted physically as adding absorption in the original model. For the standard Helmholtz equation, or for the convected Helmholtz equation in a subsonic flow, coercivity is recovered for any $\varepsilon > 0$, even arbitrarily small. This will not ne the case here for the Goldstein's problem, due to the coupling with the transport equation: we shall need the absorption $\varepsilon$ to be large enough. This will appear as a technical necessity but this is also physically meaningful: in aeroacoustics the possible presence of hydrodynamic instabilities require a large enough absorption to be counterbalanced.

**Proposition 6.** For a strictly subsonic $\Omega$-filling flow $v_0$, and for $\omega_* = i\omega_i$ with $\omega_i > 0$ large enough (obviously $\omega_* \in C_\beta^+$), $A^g(\omega_*)$ is invertible.

**Proof.** Of course, it suffices to prove the coercivity of $a^g(i\omega_i, \cdot, \cdot)$ for $\omega_i$ large enough. Let us begin with $a(i\omega; \varphi, \varphi)$ (associated to the convected Helmholtz's problem). Noticing that

$$ \Ree [ \varphi (M_0 \cdot \nabla \bar{\varphi}) - (M_0 \cdot \nabla \varphi) \bar{\varphi} ] = 0, $$

we see that

$$ \Ree a(i\omega_i; \varphi, \varphi) = \int_{\Omega} \rho_0 (|\nabla\varphi|^2 - |M_0 \cdot \nabla\varphi|^2 + \frac{\omega_i^2}{c_0^2} |\varphi|^2) + \omega_i \int_{\partial\Omega} \rho_0 c_0^{-1} \lambda |\varphi|^2. $$

Then, using $\lambda \ge 0$ on $\partial\Omega$, one gets:

$$ \Ree [a(i\omega_i; \varphi, \varphi)] \ge \alpha_*(\omega_i) \|\varphi\|_{H^1(\Omega, \rho_0)}^2, \quad \alpha_*(\omega_i) := \min \left(1 - \|M_0\|_{L^\infty}^2, \omega_i^2 \|c_0\|_{L^\infty}^{-2}\right). \quad (54) $$

Next, we treat the remaining part (50) of $a^g$, namely (see (50))

$$ \Ree \left( d \left( \mathbf{S}_0(i\omega_i; \nabla\varphi), \psi \right) \right) = \Ree \left( \int_{\Omega} \rho_0 \mathbf{S}_0(i\omega_i; \nabla\varphi) \cdot \nabla\bar{\varphi} \right) - \Ree \left( \int_{\Gamma_+} \rho_0 (\mathbf{M}_0 \cdot \mathbf{n})(\mathbf{S}_0(i\omega_i; \nabla\varphi) \cdot \mathbf{n}) \bar{\varphi} \right). $$

First, by Cauchy-Schwarz,

$$ \left| \int_{\Omega} \rho_0 \mathbf{S}_0(i\omega_i; \nabla\varphi) \cdot \nabla\bar{\varphi} \right| \le \|\mathbf{S}_0(i\omega_i; \nabla\varphi)\|_{L^2(\Omega, \rho_0)^d} \|\nabla\varphi\|_{L^2(\Omega, \rho_0)^d}, $$

so that, using (48)(i),

$$ \left| \int_{\Omega} \rho_0 \mathbf{S}_0(i\omega_i; \nabla\varphi) \cdot \nabla\bar{\varphi} \right| \le C_0(v_0, \Omega, i\omega_i) \|\omega_0\|_{L^\infty} \|\nabla\varphi\|_{L^2(\Omega, \rho_0)^d}^2. \quad (55) $$
In a similar way, using again Cauchy-Schwarz, $M_0 = c_0^{-1} v_0$ and the boundedness of $\rho_0$ and $c_0^{-1}$, we get:

$$ \left| \int_{\Gamma_+} \rho_0 (M_0 \cdot n) (S_0(i\omega_i; \nabla\varphi) \cdot n) \bar{\varphi} \right| \le \|c_0^{-1}\|_{L^\infty} \|(S_0(i\omega_i; \nabla\varphi) \cdot n)\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))} \|\varphi\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}. $$

Thus, introducing the non dimensionless constant

$$ C_{tr}(\rho_0, v_0, \Omega) := \sup_{\varphi \in H^1(\Omega)} \frac{\|\varphi\|_{L^2(\Gamma_+, \rho_0(v_0 \cdot n))}}{\|\varphi\|_{H^1(\Omega, \rho_0)}} < +\infty \quad (\text{by trace theorem}), $$

and using (48)(ii), we conclude that

$$ \left| \int_{\Gamma_+} \rho_0 (M_0 \cdot n) (S_0(\omega; \nabla\varphi) \cdot n) \bar{\varphi} \right| \le \|c_0^{-1}\|_{L^\infty} \|\omega_0\|_{L^\infty} C_{tr}(\rho_0, v_0) C_+(v_0, \Omega, \omega_i) \|\varphi\|_{H^1(\Omega, \rho_0)}^2. \quad (56) $$

Finally, regrouping (54), (55), (56) in (50) and setting

$$ \alpha(\omega_i) := \alpha_*(\omega_i) - [C_0(v_0, \Omega, \omega_i) + \|c_0^{-1}\|_{L^\infty} C_{tr}(\rho_0, v_0) C_+(v_0, \Omega, \omega_i)] \|\omega_0\|_{L^\infty(\Omega)^2}, $$

we obtain the lower bound

$$ \Re e \left( a^g(i\omega_i; \varphi, \varphi) \right) \ge \alpha(\omega_i) \|\varphi\|_{H^1(\Omega, \rho_0)}^2. $$

As $C_0(v_0, \Omega, i\omega_i)$ and $C_1(v_0, \Omega, i\omega_i)$ tends to 0 when $\omega_i \to +\infty$, cf. Remark 10, and since $\alpha_*(\omega_i) = 1 - \|M_0\|_{L^\infty}^2$, for $\omega_i$ large enough (see (54)), we have

$$ \lim_{\omega_i \to +\infty} \alpha(i\omega_i) = 1 - \|M_0\|_{L^\infty}^2 > 0. \quad (57) $$

It is then easy to conclude.

<figure><img src="image_6.png" /></figure>

**Remark 12.** It is worthwhile noticing that Proposition 6 does not require that the flow is admissible. Only the $\Omega$-filling property, for defining $S_0(i\omega_i; \nabla\varphi)$, and the fact that the flow is subsonic, for (57), are needed. This is explained by the fact that the hydrodynamic effects, due to the unknown $\xi$, are killed at high absorption as shown by Theorem 4.

End of the proof of the main result Theorem 1. Since we have gathered all the Corollary 1 conditions, there exists a discrete subset $\mathcal{R}_{ex}^c \subset \mathbb{C}_\beta^+$, such that for all frequency $\omega \in \mathbb{C}_\beta^+ \setminus \mathcal{R}_{ex}^c$, the reduced Goldstein's problem (24) is well-posed and so is the Goldstein's problem (14, 15). We deduce that the same is true for all frequencies $\omega \in \mathbb{R} \setminus \mathcal{R}_{ex}$ where the set $\mathcal{R}_{ex} := \mathcal{R}_{ex}^c \cap \mathbb{R}$ is discrete, which ends the proof of the main result.

### 3.5. On the admissibility condition as a low vorticity condition

In this section, essentially for technical reasons, we suppose that $d = 2$ that allows us to use the simple characterization (19) of $\Omega$-filling flows. However, we conjecture that the content of this section remains valid for $d = 3$.

We wish to reinterpret the condition (21) as imposing on the mean flow to be subsonic, as in the potential case, but also to be of low vorticity and not more. In this aim, we consider a family of
flows parameterized by a real $\eta \in \mathbb{R}$, ($\rho_0^\eta, p_0^\eta, v_0^\eta$) whose velocity flows are perturbation of a (strictly) subsonic potential flow. More precisely, these are of the form:

$$v_0^\eta := v_0 + \eta w_0, \quad v_0 = \nabla\varphi_0, \quad \|\nabla \times w_0\|_\infty = 1, \qquad (58)$$

and we assume that the acoustic velocity field $c_0^\eta$ satisfies

$$\lim_{\eta \to 0} \|c_0^\eta - c_0\|_{L^\infty} = 0.$$

The vorticity of these flows is proportional to $\eta$ (in such a way that, for these flows “small vorticity” is equivalent to “small $\eta$”):

$$\omega_0^\eta := \nabla \times (\nabla\varphi_0 + \eta w_0) = \eta \nabla \times w_0 \implies \|\omega_0^\eta\|_\infty = |\eta|.$$

We also assume that $w_0$ satisfies

$$w_0 \cdot n = 0 \quad \text{on } \partial\Omega, \qquad (59)$$

(which is for instance the case if $\text{supp}(w_0) \subset \Omega$), such that the inflow and outflow boundaries (cf. definition 2) are independent of $\eta$, i.e., with the notations of Remark 2:

$$\forall \eta, \quad \Gamma_\pm(v_0^\eta) = \Gamma_\pm(v_0) \equiv \Gamma_\pm.$$

**Remark 13.** We think that the assumption is not really essential but it suppresses many tedious difficulties in the proof of our results.

**Theorem 5.** Let $v_0$ strictly be subsonic and $\Omega$ filling and $v_0^\eta$ be defined by (58) where the vector field $w_0$ satisfies (59). Then, there is a threshold $\eta^* > 0$ such that for all $\eta \in ]-\eta^*, \eta^*[$, the flow $v_0^\eta$ is admissible in the sense of Definition 2 and, therefore, so that the Goldstein's problem (14, 15) associated with the flow ($\rho_0^\eta, p_0^\eta, v_0^\eta$) is well posed in the sense of Theorem 1.

The proof relies on the following technical lemma (whose proof is given in appendix ) that provides a uniform control (in $\eta$) of the life times of the flows $v_0^\eta$

**Lemma 2.** There exists $\eta^* > 0$ and $T^* > 0$ such that :

$$\forall \eta \in (-\eta^*, \eta^*), \quad t^*(v_0^\eta, \Omega) \le T^*.$$

**Proof of Theorem 5.** One first remarks that, using the 2D characterization (19) of the $\Omega$-filling property, for $\eta$ small enough, as $\inf_\Omega |v_0^\eta| > 0$, $v_0^\eta$ is $\Omega$-filling.

Then, if we denote $M_0^\eta := v_0^\eta/c_0^\eta$ the Mach flow of the perturbed flow, with our assumptions, $M_0^\eta$ converges uniformly in $\Omega$ towards $M_0^\eta := v_0^\eta/c_0^\eta$ and in particular $\|M_0^\eta\|_{L^\infty} \to \|M_0\|_{L^\infty}$ when $\eta \to 0$. In a same fashion, one easily sees that $\|\nabla v_0^\eta\|_{L^\infty} \to \|\nabla v_0\|_{L^\infty}$ when $\eta \to 0$. Finally, as a consequence of Lemma 2, as $t^*(v_0^\eta, \Omega)\Phi(2t^*(v_0^\eta, \Omega)\|\nabla v_0^\eta\|_{L^\infty})$ remains bounded when $\eta \to 0$ and as $\|\omega_0^\eta\|_{L^\infty} = |\eta| \to 0$, one gets:

$$\lim_{\eta \to 0} \left[ 1 - \|M_0^\eta\|_{L^\infty}^2 - \|\omega_0^\eta\|_\infty t^*(v_0^\eta, \Omega) \Phi(2t^*(v_0^\eta, \Omega)\|\nabla v_0^\eta\|_{L^\infty}) \right] = 1 - \|M_0^\eta\|_{L^\infty}^2 > 0.$$

Thus $v_0^\eta$ is admissible for $\eta$ small enough. One concludes with Theorem 1.

□
APPENDIX A. A FIRST ORDER ABSORBING BOUNDARY CONDITION

In this appendix, we explain the construction of the absorbing boundary condition (11) of Section 2.3.1, as an approximate transparent condition.

In what follows, we denote $\Omega_{ext}$ the exterior domain as the unbounded connected component of $\mathbb{R}^d \setminus \Gamma$, so that in particular $\Gamma = \partial\Omega_{ext}$. Ideally, a transparent boundary condition on $\partial\Omega$ would result from a generalized Dirichlet-to-Neumann (DtN) operator acting on a couple of boundary data

$$T : (\varphi^0, \boldsymbol{\xi}^0) : \Gamma \times \Gamma_+ \to \mathbb{C} \times \mathbb{C}^d,$$

and is related to the notion of generalized Neumann trace defined in (16):

$$T(\varphi^0, \boldsymbol{\xi}^0) := -(\nabla\phi(\varphi^0, \boldsymbol{\xi}^0) + \zeta(\varphi^0, \boldsymbol{\xi}^0)) \cdot \boldsymbol{n} + c_0^{-1}(M_0 \cdot \boldsymbol{n})D_\omega\phi(\varphi^0, \boldsymbol{\xi}^0), \quad (60)$$

where $\phi := \phi(\varphi^0, \boldsymbol{\xi}^0)$ and $\zeta := \zeta(\varphi^0, \boldsymbol{\xi}^0)$ are solution of the exterior Dirichlet boundary value problem (posed in $\Omega_{ext}$) associated to the convected Helmholtz equation with constant coefficients ($c_0, M_0$) (the exterior medium is homogeneous with a uniform flow):

$$\left\{
\begin{array}{ll}
\left(-i \displaystyle\frac{\omega}{c_0} + M_0 \cdot \nabla\right)^2 \phi - \Delta\phi - \nabla \cdot \zeta = 0, & \text{in } \Omega_{ext} \\[2ex]
\left(-i \displaystyle\frac{\omega}{c_0} + M_0 \cdot \nabla\right) \zeta = 0, & \text{in } \Omega_{ext} \\[2ex]
\phi = \varphi_0 \quad \text{on } \Gamma, \quad \zeta = \xi_0 \quad \text{on } \Gamma_+.
\end{array}
\right.
\qquad (61)$$

Note that the outflow boundary $\Gamma_+$ becomes the inflow boundary for the exterior domain reason why we need to prescribe $\zeta$ on $\Gamma_+$ (instead on $\Gamma_+$ for the interior problem, cf. Section 2.3). Also note that the transport equation for $\zeta$ is completely decoupled from $\phi$ because the vorticity $\omega_0$ is 0 in the exterior domain $\Omega_{ext}$.

Once $T$ is known, a transparent boundary condition for the interior problem in $(\varphi, \boldsymbol{\xi})$ is

$$\rho_0(\nabla\varphi + \boldsymbol{\xi}) \cdot \boldsymbol{n} - \rho_0 c_0^{-1}(M_0 \cdot \boldsymbol{n})D_\omega\varphi + \rho_0 T(\varphi, \boldsymbol{\xi}) = 0 \quad \text{on } \Gamma. \qquad (62)$$

The above operator $T$ in non local along $\Gamma$ and can not be computed in practice. That is why or goal in this section will be to construct a “local approximation” based on the same “idea” that for the scalar wave equation for instance. This idea consists in considering the problem in a small neighborhood of a point $x \in \Gamma$ with outgoing normal vector $n$ and

* assimilate locally the boundary $\Gamma$ to the tangent plane $\Gamma_x$ and the exterior domain to the half-space $\Omega_x := \{y / (y - x) \cdot n > 0\}$,

* consider that around $\Gamma$, the boundary data are constant (in other words their lateral variations are neglected).

This leads to consider the ($x$-dependent) half-space problem (61) in which $\Omega_{ext}$ is replaced by $\Omega_x$ and the boundary data are constant along $\Gamma_x$. i.e. $\varphi_0 \equiv \varphi_0(x)$ and $\xi_0 \equiv \xi_0(x)$. By translational invariance along $\Gamma_x$ we deduce that the solution $(\phi_x, \zeta_x)$ is a 1D function of the space coordinate
normal to $\Gamma_x$. If we call $y$ this variable, so that $\Omega_x$ corresponds to $y = 0$ and $\Gamma_x$ corresponds to $y = 0$, the 1D problems rewrites

$$ egin{cases} 
\left(-i \frac{\omega}{c_0} + (M_0 \cdot n) \frac{d}{dy}\right)^2 \phi_x - \frac{d^2 \phi_x}{dy^2} - \frac{d}{dy}(\zeta_x \cdot n) = 0, & y > 0, \\ 
\left(-i \frac{\omega}{c_0} + (M_0 \cdot n) \frac{d}{dy}\right) \zeta_x = 0, & y > 0, \\ 
\phi_x(0) = \varphi_0(x) \quad \text{on } \Gamma, \quad \zeta_x(0) = \xi_0(x) \quad \text{on } \Gamma_+. 
\end{cases} 
\quad (63) $$

Note that the above is valid for any point $x$ along the boundary $\Gamma$ with the particularity that $\xi_0(x) = 0$ if $x \notin \Gamma_+$. From the transport equation we deduce that

$$ \zeta_x(y) = \xi_0(x) e^{i \frac{\omega}{c_0} \frac{y}{M_0 \cdot n}} \quad (64) $$

that we substitute into the equation for $\phi_x$, which gives

$$ \left(-i \frac{\omega}{c_0} + (M_0 \cdot n) \frac{d}{dy}\right)^2 \phi_x - \frac{d^2 \phi_x}{dy^2} = i \frac{\omega}{c_0} \left(\frac{\xi_0(x) \cdot n}{M_0 \cdot n}\right) e^{i \frac{\omega}{c_0} \frac{y}{M_0 \cdot n}}. $$

The solution is the sum of a particular solution $\phi_p + \phi_h$ where $\phi_h$ is a solution of the homogeneous equation. Of course $\phi_p$ is of the form

$$ \phi_p(y) = \phi_{0,p} e^{i \frac{\omega}{c_0} \frac{y}{M_0 \cdot n}}, $$

where, noticing that the first term of the left hand side of the previous equation vanishes for $\phi_x = \phi_p$, one computes that

$$ \phi_{0,p} = i \frac{c_0}{\omega} (M_0 \cdot n) (\xi_0(x) \cdot n). $$

On the other hand, we know that $\phi_h$ is of the form

$$ \phi_h(y) = A^+ e^{ik^+(\omega)y} + A^- e^{ik^-(\omega)y}, $$

where $(k^+(\omega), k^-(\omega))$ are the two solutions of the quadratic dispersion relation in $k$:

$$ k^2 = \left( \frac{\omega}{c_0} - (M_0 \cdot n) k \right)^2, $$

that is to say, taking into account that the flow is subsonic ($|M_0| < 1$),

$$ k^+(\omega) = \frac{\omega}{c_0} (1 + M_0 \cdot n)^{-1} > 0, \quad k^-(\omega) = -\frac{\omega}{c_0} (1 - M_0 \cdot n)^{-1} < 0. $$

Moreover, we shall retain only in the homogeneous solution, the one who is going at infinity (this can be fully justified by limiting absorption, we omit the details) which yields $A^- = 0$. Therefore $A^+ = \varphi_0(x) - \phi_{0,p}$ in order that $\phi_x(0) = \varphi_0(x)$. This leads to

$$ \phi_x(y) = \varphi_0(x) e^{i \frac{\omega}{c_0} \frac{x}{1+M_0 \cdot n}} + i \frac{c_0}{\omega} (M_0 \cdot n) (\xi_0(x) \cdot n) \left[ e^{i \frac{\omega}{c_0} \frac{y}{M_0 \cdot n}} - e^{i \frac{\omega}{c_0} \frac{y}{1+M_0 \cdot n}} \right]. \quad (65) $$
We then define the approximate operator, consistently with the 1D approximate solution $(\phi_x, \zeta^x)$
((60)), as

$$
\begin{align}
-T_{ap}(\varphi_0, \boldsymbol{\xi}_0)(\boldsymbol{x}) &= \phi_x'(0) + \zeta^x(0) \cdot \boldsymbol{n} + (M_0 \cdot \boldsymbol{n})(i\omega c_0^{-1}\phi_x - (M_0 \cdot \boldsymbol{n})\phi_x')(0) \tag{66} \\
&= [1 - (M_0 \cdot \boldsymbol{n})^2]\phi_x'(0) + i\omega c_0^{-1}(M_0 \cdot \boldsymbol{n})\phi_x(0) + \zeta^x(0) \cdot \boldsymbol{n} \nonumber
\end{align}
$$

By (63), $\phi_x(0) = \varphi_0(x)$ and $\zeta^x(0) = \xi_0(x)$, thus

$$
i\omega c_0^{-1} (M_0 \cdot n) \phi_x(0) + \zeta^x(0) \cdot n = i\omega c_0^{-1} (M_0 \cdot n) \varphi_0(x) + \xi_0(x) \cdot n, \quad (67)
$$

while, from (65), one gets $(\phi_x)'(0) = (1 + M_0 \cdot n)^{-1} (i \omega c_0^{-1} \varphi_0(x) - \xi_0(x) \cdot n)$, thus

$$
[1 - (M_0 \cdot n)^2](\phi_x)'(0) = (1 - M_0 \cdot n) (i \omega c_0^{-1} \varphi_0(x) - \xi_0(x) \cdot n). \quad (68)
$$

Then, substituting (67) and (68) into (66) gives which, using (64, 65), gives

$$
T_{ap}(\varphi_0, \boldsymbol{\xi}_0)(\boldsymbol{x}) = -i \frac{\omega}{c_0} \varphi_0(\boldsymbol{x}) - \boldsymbol{\xi}_0(\boldsymbol{x}) \cdot \boldsymbol{n}.
$$

Finally, changing $T$ into $T_{ap}$ in the transparent condition (62) leads to the desired absorbing boundary condition (11).

APPENDIX B. PROOF OF THE TECHNICAL LEMMA 2

Let us first recall the statement of this lemma:

**Lemma 3.** *There exists $\eta^* > 0$ and $T^* > 0$ such that:*

$$
\forall \eta \in (-\eta^*, \eta^*), \quad t^*(v_0^\eta, \Omega) \le T^*.
$$

**Proof.** In the same fashion that $\chi$ denotes the characteristics of the flow $v_0$, let us introduce
$\chi^\eta$ the characteristics of the perturbed flow $v_0^\eta$, i.e. $\chi^\eta : \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}^d$ solution of the following
system of ordinary differential equations:

$$
\left\{
\begin{array}{ll}
\partial_t \chi^\eta(t, \boldsymbol{x}) = \boldsymbol{v}_0^\eta(\chi^\eta(t, \boldsymbol{x})), & \text{for all } (t, \boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d. \\
\chi^\eta(0, \boldsymbol{x}) = \boldsymbol{x}, &
\end{array}
\right.
$$

The proof consists of the following steps for which a justification will be given:

(1) General theorems of ODE theory provide that the mappings $(t, \boldsymbol{x}) \mapsto \chi^\eta(t, \boldsymbol{x})$ converge uniformly on any compact of $\mathbb{R} \times \mathbb{R}^d$ when $\eta \to 0$. However, in our case, we can merely get a stronger result by establishing the following estimate:

$$
\begin{gather*}
\forall \eta \in \mathbb{R}, \forall (t, \boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d, \\
|\chi^\eta(t, \boldsymbol{x}) - \chi(t, \boldsymbol{x})| \le |\eta| (t + t^2 \|\nabla v_0\|_{L^\infty} e^{t \|\nabla v_0\|_{L^\infty}}) \|w_0\|_{L^\infty}.
\end{gather*}
$$

(2) There exists a small enough $\varepsilon > 0$ such that the flow $v_0$ still fills the extended domain $\Omega^\varepsilon := \Omega + B(0, \varepsilon)$ (i.e. $v_0$ is $\Omega^\varepsilon$-filling). Thus, the exit time of $\Omega^\varepsilon$ for the flow $v_0$ is well-defined by:

$$
t^*(v_0, \Omega^\epsilon) := \sup_{x \in \Omega^\epsilon} t^*(x, v_0, \Omega^\epsilon) < +\infty,
$$
where the exit time starting from $x \in \Omega^\varepsilon$ is defined by:

$$t^*(x, v_0, \Omega^\varepsilon) := \sup \{t \ge 0, / \quad \forall \tau \in [0, t), \chi(t, x) \in \Omega^\varepsilon\}.$$

(3) There exists $\eta^* > 0$ such that:

$$\forall \eta \in (-\eta^*, \eta^*), \quad t^*(v_0^\eta, \Omega) \le t^*(v_0, \Omega^\varepsilon),$$

which provides the expected bound with $T^* := t^*(v_0, \Omega^\varepsilon)$.

Justifications of the steps (1), (2) and (3).

(1) The inequality follows a Gronwall estimation of the difference $\chi^\eta(t, x) - \chi(t, x)$ using by using the ODE and the fact that $v_0^\eta := v_0 + \eta w_0$. More precisely, for $(t, x) \in \mathbb{R}^+ \times \mathbb{R}^d$, we have:

$$\begin{align*}
\chi^\eta(t, x) - \chi(t, x) &= \chi^\eta(0, x) - \chi(0, x) + \int_0^t \partial_t \chi^\eta(s, x) - \partial_t \chi(s, x) \mathrm{d}s \\
&= \int_0^t v_0^\eta(\chi^\eta(s, x)) - v_0(\chi(s, x)) \mathrm{d}s \\
&= \int_0^t v_0^\eta(\chi^\eta(s, x)) - v_0(\chi^\eta(s, x)) + v_0(\chi^\eta(s, x)) - v_0(\chi(s, x)) \mathrm{d}s \\
&= \int_0^t \eta w_0(\chi(s, x)) + v_0(\chi^\eta(s, x)) - v_0(\chi(s, x)) \mathrm{d}s.
\end{align*}$$

Then, by mean value inequality,

$$|\chi^\eta(t, x) - \chi(t, x)| \le t |\eta| \|w_0\|_{L^\infty} + \|\nabla v_0\|_{L^\infty} \int_0^t |\chi^\eta(s, x) - \chi(s, x)| \mathrm{d}s,$$

and by Gronwall's lemma we deduce

$$\begin{align*}
|\chi^\eta(t, x) - \chi(t, x)| &\le t |\eta| \|w_0\|_{L^\infty} + \|\nabla v_0\|_{L^\infty} \int_0^t e^{(t-s)\|\nabla v_0\|_{L^\infty}} s |\eta| \|w_0\|_{L^\infty} \mathrm{d}s, \\
&\le t |\eta| \|w_0\|_{L^\infty} + \|\nabla v_0\|_{L^\infty} e^{t\|\nabla v_0\|_{L^\infty}} t^2 |\eta| \|w_0\|_{L^\infty},
\end{align*}$$

which is the expected inequality.

(2) The 2D characterization (19) of $\Omega$-filling states that $v_0$ is $\Omega$-filling if and only $\inf_\Omega |v_0| > 0$. Thus, it is straightforward that the continuity of $v_0$ and $\inf_\Omega |v_0| > 0$ imply that there exists $\varepsilon > 0$ such that $\inf_{\Omega^\varepsilon} |v_0| > 0$.

(3) Thanks to 1., let $\eta^* > 0$ such that

$$\forall \eta \in (-\eta^*, \eta^*), \quad \forall (t, x) \in [0, t^*(v_0, \Omega^\varepsilon)] \times \Omega^\varepsilon, \quad |\chi^\eta(t, x)) - \chi(t, x)| \le \varepsilon/2.$$

Then, let $x \in \Omega$. One has $\chi(t^*(x, v_0, \Omega^\varepsilon), x) \in \partial\Omega^\varepsilon$ and thus is at a distance $\varepsilon$ to $\Omega$:

$$\operatorname{dist} (\chi(t^*(x, v_0, \Omega^\varepsilon), \Omega), \Omega) = \varepsilon.$$

One also has that for $\eta \in (-\eta^*, \eta^*)$, $(t^*(x, v_0, \Omega^\varepsilon), x) \in [0, t^*(v_0, \Omega^\varepsilon)] \times \Omega^\varepsilon$, leading to $|\chi^\eta(t^*(x, v_0, \Omega^\varepsilon), x) - \chi(t^*(x, v_0, \Omega^\varepsilon), x)| \le \varepsilon/2$, which implies that

$$\chi^\eta(t^*(x, v_0, \Omega^\varepsilon), x) \notin \Omega.$$
By definition of $t^*(x, v_0^\eta, \Omega)$, we have $\forall \tau \in [0, t^*(x, v_0^\eta, \Omega))$, $\chi^\eta(\tau, x) \in \Omega$, from which we deduce:

$$t^*(x, v_0^\eta, \Omega) \le t^*(x, v_0, \Omega^\epsilon) \le t^*(v_0, \Omega^\epsilon).$$

Passing to the sup over $\Omega$ in this inequality, one finally gets, for any $\eta \in (-\eta^*, \eta^*)$:

$$t^*(v_0^\eta, \Omega) \le t^*(v_0, \Omega^\epsilon),$$

which is the announced result.

<figure><img src="image_7.png" /></figure>

## APPENDIX C. ON AN ALTERNATIVE APPROACH TO THE EXISTENCE RESULT

Let us mention a possible alternative to the approach of Section 3 to the analysis of Goldstein's coupling. We have not chosen to follow this approach in a first step for reasons that we shall mention later, but this could be the object of a companion paper.

Let us restart from the abstract block form (32) of the Goldstein's problem:

$$\begin{pmatrix} A(\omega) & D \\ \mathbf{B} & T(\omega) \end{pmatrix} \begin{pmatrix} \varphi \\ \xi \end{pmatrix} = \begin{pmatrix} \hat{f} \\ \mathbf{0} \end{pmatrix}.$$

In our approach in Section 3.1, we proceed via the elimination of $\xi$. An natural alternative would be to eliminate $\varphi$ via (formally)

$$\varphi = A(\omega)^{-1}(\hat{f} - D\xi),$$

so that, we are led to a "reduced" transport equation on $\xi$

$$T(\omega) - \mathbf{B} A(\omega)^{-1} D = - \mathbf{B} A(\omega)^{-1} \hat{f}. \qquad (69)$$

To do so, we first have to check that the first step is possible, which is related to the invertibility of $\mathbf{A}(\omega)$, that is to say the resolution of the convected Helmholtz equation. We know, see Section 3.3, that $\mathbf{A}(\omega)$ is invertible under the only assumption that the flow is strictly subsonic. Then it remains to treat the reduced transport equation which is far from standard since the operator $\mathbf{B} A(\omega)^{-1} D$ is a fully non local perturbation of the transport operator.

pouet For the perturbation analysis, we can not use, as in Section 3.4, the Fredholm approach which is not adapted (at least to our knowledge) to the transport equation. However, we can try to use the Banach fixed point theorem. More precisely, assuming that the transport operator $T(\omega)$ is invertible, which is guaranteed (Section 3.4.2) under the condition that the flow is $\Omega$-filling, we rewrite (69) as

$$I - T(\omega)^{-1} \mathbf{B} A(\omega)^{-1} D = - T(\omega)^{-1} \mathbf{B} A(\omega)^{-1} \hat{f}.$$

We can then conclude to the solvability of (69) under the formal condition

$$\|T(\omega)^{-1} \mathbf{B} A(\omega)^{-1} D\| < 1, \qquad (70)$$

where $T(\omega)^{-1} \mathbf{B} A(\omega)^{-1} D$ is seen as an operator of $\mathcal{L}(\mathbf{M})$. Of course a sufficient condition for (70) is

$$\|\mathbf{B}\| \|D\| \|T(\omega)^{-1}\| \|A(\omega)^{-1}\| < 1. \qquad (71)$$

If, for a while, we forget about the presence of $\|A(\omega)^{-1}\|$, considering the estimate (48)(i) (remember that $T(\omega)^{-1} = \mathbf{S}_0(\omega; \nabla\varphi)$, one sees that the condition (71) is at least qualitatively very similar
to our admissibility condition (2), the norm $||\omega_0||_{L^\infty}$ of the vorticity being hidden in the estimate (48)(i) which is frequency independent for real frequencies. However, the presence of $||A(\omega)^{-1}||$ makes the condition (71) much less explicit than (2), especially because it is hard to get an explicit upper bound for $||A(\omega)^{-1}||$. This quantity does depend on the frequency and may become large if the presence of resonances of the convected Helmholtz equation (i.e. poles of the resolvent $A(\omega)^{-1}$) that could be close to the real axis. This is one of the reasons that led us to privilege the approach adopted in this paper.

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