Models of Nonlinear Acoustics Viewed as Approximations of the Kuznetsov Equation

We relate together different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the L 2 norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

One of the most general model to describe an acoustic wave propagation in an homogeneous thermo-elastic medium is the compressible Navier-Stokes system in R n ∂ t ρ + div(ρv) = 0, ρT where the pressure p is given by the state law p = p(ρ, S) . The density ρ , the velocity v , the temperature T and the entropy S are unknown functions in system (1)- (4). The coefficients ζ, κ and η are constant viscosity coefficients. For the acoustical framework the wave motion is supposed to be potential and the viscosity coefficients are supposed to be small in terms of a dimensionless small parameter ε > 0 , which also characterizes the size of the perturbations near the constant state (ρ 0 , 0, S 0 , T 0 ) . Here the velocity v 0 is taken equal to 0 just using a Galilean transformation. Actually, ε is the Mach number, which is supposed to be small [5] ( ǫ = 10 −5 for the propagation in water with an initial power of the order of 0.3 W/cm 2 ): where c = p ′ (ρ 0 ) is the speed of sound in the unperturbed media. Hence as in [12,43], system (1)-(4) becomes an isentropic Navier-Stokes system with the approximate state equation p(ρ, S) = p(ρ ε ) + O(ε 3 ) : where γ = C p /C V denotes the ratio of the heat capacities at constant pressure and at constant volume respectively and with a small enough and positive viscosity coefficient: If we go on physical assumptions of the wave motion [5,18,31,50] for the perturbations of the density or of the velocity or of the pressure, the isentropic system (5)-(6) gives 1. the Westervelt equation for the potential of the velocity, derived initially by Westervelt [50] and later by other authors [1,49]: with the same constants introduced for the Navier-Stokes system.
2. the Kuznetsov equation also for the potential of the velocity, firstly introduced by Kuznetsov [31] for the velocity potential, see also Refs. [18,23,28,33] for other different methods of its derivation: For higher order models as the nonlinear Jordan-Moore-Gibson-Thompson (JMGT) equation, containing the Kuznetsov equation as a particular or a limit case, see [24,26,27] and their references. In this article we don't consider such higher order models and focus our attention on the Kuznetsov equation considered here as the most complete equation.
In [12] it is shown that the Kuznetsov equation comes from the Navier-Stokes or Euler system only by small perturbations, but to obtain the KZK and the NPE equations we also need to perform in addition to the small perturbations a paraxial change of variables. In this article we derive the KZK and the NPE equations from the Kuznetsov equation just performing the corresponding paraxial change of variables and show that the Westervelt equation can be also viewed as an approximation of the Kuznetsov equation by a nonlinear perturbation.
The physical context and the physical usage of the KZK and the NPE equations are different: the NPE equation is helpful to describe short-time pulses and a long-range propagation, for instance, in an ocean wave-guide, where the refraction phenomena are important [7,38], while the KZK equation typically models the ultrasonic propagation with strong diffraction phenomena, combining with finite amplitude effects (see [42] and the references therein). But in the same time [12], there is a bijection between the variables of these two models and they can be presented by the same type differential operator with constant positive coefficients: x ± c 3 ∆ y for t ∈ R + , x ∈ R, y ∈ R n−1 .
Therefore, the results on the solutions of the KZK equation from [21,41] are valid for the NPE equation. The interest to study how closed are the solutions of the general model of the nonlinear wave motion, described by the Kuznetsov equation, and of simplified models with more particular area of application (such as the KZK equation and the NPE equation which are valid only with additional assumptions on the wave propagation describing by the paraxial changes of variables) is naturally motivated by the questions about the accuracy of the approximations and of a comparative analysis of the solutions of these models.
If we formally consider the differential operators in the Kuznetsov equation (9) and the Westervelt equation (8), we notice that the Westervelt equation keeps only one of two non-linear terms of the Kuznetsov equation, producing cumulative effects in a progressive wave propagation [1]. The question how closed are the solutions of these two models, which differ on presence of local nonlinear term, was also open so far.
The mentioned approximation questions are treated theoretically in this article. Let us also notice that the Kuznetsov equation (9) (and also the Westervelt equation (8)) is a non-linear wave equation with terms of different order (the wave operator is of order ε 0 and the nonlinear and viscosity terms are of order ε ). But the KZK-and NPE-paraxial approximations allow to have the approximate equations with all terms of the same order, i.e. the KZK and NPE equations. For the well posedness of the Cauchy problem for the Kuznetsov equation we cite [11] and for boundary value problems in regular bounded domains see [25,29,40].
We present the structure of the paper and its mains results in the next subsection.

Main results
To keep a physical sense of the approximation problems, we consider especially the two or three dimensional cases, i.e. R n with n = 2 or 3 , and in the following we use the notation x = (x 1 , x ′ ) ∈ R n with one propagative axis x 1 ∈ R and the traversal variable x ′ ∈ R n−1 .
To be able to consider the approximation of the Kuznetsov equation by the KZK equation (see Section 2), we establish (see Theorems 7 and 8 in Appendix A) global well posedness results for the Kuznetsov equation in the half space similar to the previous framework for the KZK and the Navier-Stokes system considered in [12]. Theorem 7 corresponds to the well posedness of the periodic in time Dirichlet boundary valued problem for the Kuznetsov equation in the half space R + × R n−1 (see Eq. (21)) for small enough boundary data. In this case the boundary condition is considered as the initial condition of the corresponding Cauchy problem in R n . The proof is based on the maximal regularity result for the corresponding linear problem given in Theorem 6 and on the application of a result of the nonlinear functional analysis from [48,1.5. Cor.,p. 368] (see also [11,Thm. 4.2]). We also applied it to prove (see Theorem 8) the well posedness for the initial boundary valued problem for the Kuznetsov equation in the half space (28), once again combining with the maximal regularity result for the linear problem (see the proof of Lemma 2).  Theorem  Theorem 1  Theorem 2  Theorem 3  Theorem 5 Derivation paraxial approximation In Subsection 2.1 we derive the KZK equation from the Kuznetsov equation by introducing the paraxial change of variables (34).
For the approximation framework for their solutions we study two cases. The first case is treated in Sub-subsection 2.2.1 and it considers the purely time periodic boundary problem in the ansatz variables (z, τ, y) moving with the wave, where we use the wellposedness result of Theorem 7. In this case the only viscous medium can be considered as the condition to be periodic in time is not compatible with shock formations providing the loss of the regularity which may occur in the inviscid medium (see [41,Thm. 1.3]). The approximation results are formulated in Theorem 1.
The second case (see Sub-subsection 2.2.2) studies the initial boundary-value problem for the Kuznetsov equation  We denote by u a solution of the "exact" problem for the Kuznetsov equation Exact(u) = 0 and by u an approximate solution, constructed by the derivation ansatz from a regular solution of one of the approximate models (for instance of the KZK or of the NPE equations), i.e. u is a function which solves the Kuznetsov equation up to ε terms, denoted by εR : In the approximation between the solutions of the Kuznetsov equation and of the Westervelt equation the remainder term appears with the size ε 2 (it is natural since both models contain terms of order ε 0 and ε ). We can summarize the obtained approximation results of the Kuznetsov equation in the following way: if, once again, u is a solution of the Kuznetsov equation and u is a solution of the NPE or of the KZK (for the initial boundary value problem) or of the Westervelt equations found for rather closed initial data then there exist constants K , C 1 , C 2 , C > 0 independent of ε , δ and on time, such that for all t ≤ C ε it holds For a more detailed comparison between different models we include the main points of our results to the comparative Table 1.
In Table 1 the line named "Initial data regularity" gives the information about the regularity of the initial data for the approximate model, which ensure the same regularity of the solutions of an approximate model and of the solution of the Kuznetsov equation, taken with the same initial data u(0) = u(0) , coming from the corresponding ansatz.
To have the remainder term R ∈ C([0, T ], L 2 (Ω)) we ensure that Exact(u) ∈ C([0, T ], L 2 (Ω)) , i.e. we need a sufficiently regular solution u . The minimal regularity of the initial data to have a such u is given in Table 1 in the last line named "Data regularity for remainder boundness".
To summarize, the rest of the paper is organized as follows. Section 2 considers the derivation (Subsection 2.   If the velocity potential is given [31] by we directly obtain from the Kuznetsov equation (9) via the paraxial change of variables that with Let us notice that the paraxial change of variables (13) defines the axis of the propagation x 1 along which the wave changes its profile much slower than along the transversal axis x ′ . This is typical for the propagation of ultrasound waves. Therefore, we find that the right-hand side ǫ -order terms in Eq. (14) is exactly the KZK equation (10). Thanks to [41, Thms. 1.1-1.3] we have the well posedness result for the KZK equation in the half space with periodic boundary conditions of a period L on τ and of mean value zero.
Due to the well posedness domain (T τ × R n−1 ) of the KZK equation, to validate the approximation between the solutions of the KZK and the Kuznetsov equations, we need to have the well posedness of the Kuznetsov equation on the half space with boundary conditions coming from the initial condition for the KZK equation. For these well posedness results see Appendix A.

Approximation of the solutions of the Kuznetsov equation by the solutions of the KZK equation.
Let us consider the Cauchy problem associated with the KZK equation  (21) with the boundary condition imposed by the initial condition I 0 of the KZK equation. In Sub-subsection 2.2.2 we study the second case, when the solution of the KZK equation, taken for τ = 0 , gives I(0, z, y) defined on R + × R n−1 , from which we deduce, according to the derivation ansatz, both an initial condition for the Kuznetsov equation at t = 0 and a corresponding boundary condition. In this second situation, it also makes sense to consider the inviscid case, briefly commented in the end of Sub-subsection 2.2.2.

Approximation problem for the Kuznetsov with periodic boundary conditions.
By [41, Thm. 1.2] there is a unique solution I(τ, z, y) of the Cauchy problem for the KZK equation (16) such that with Tτ I(l, z, y)dl = 0 and Formula (18), which implies that ∂ −1 τ I is L -periodic in τ and of mean value zero, gives us the estimate So ∂ −1 τ I| z=0 ∈ H s (Ω 1 ) , and hence by (17) with the paraxial change of variable (13) associated with the KZK equation. Thus u is L -periodic in time and of mean value zero. Now we consider the following Kuznetsov problem in which the boundary condition is imposed by the initial condition for the KZK equation: Let us define (see Eq. (12), and subsection 4.1 in [12] for more details) ThenĨ is the solution of the Kuznetsov equation written in the following form with the remainder R Kuz−KZK defined in Eq. (15): In Eq. (24) we can recognize the system associated with the KZK equation (10). Now we can formulate the following approximation result between the solutions of the KZK and Kuznetsov equations.
, there exists a unique global solution I of the Cauchy problem for the KZK equation (16) such that In addition, there exists a unique global solutionĨ of the Kuznetsov problem (24), in the senseĨ : is the global solution of the periodic problem (21) for the Kuznetsov equation with g defined by I 0 as in Eq. (22). Moreover there exist C 1 , C 2 > 0 such that and is small enough. Moreover, since g is given by Eq. (22), thanks to the definition of ∂ −1 τ in (18) and the fact that

And thus
Therefore we can use Theorem 7, which implies the global existence of the periodic in time solution of the Kuznetsov periodic boundary value problem (21) as I 0 is small enough in H s+ 3 2 (T τ × R n−1 ) . Therefore, it also implies the global existence ofĨ , defined in (23), which is the solution of the exact Kuznetsov system (24). Now we subtract the equations in systems (16) and (24) to obtain Denoting Ω 1 = T τ ×R n−1 , we multiply this equation by (I −Ĩ) , integrate over T τ ×R n−1 and perform a standard integration by parts, which gives Let us notice that This comes from the fact that in system (24) the "worst" term, asking the most regu- with a constant C > 0 independent of z thanks to (26). It leads to the estimate in which, due to the regularity of I for s and I 0 (see [41]) the term is bounded by a constant C > 0 independent of z . Consequently, we have the desired estimate and the other results follow from Gronwall's Lemma.

Approximation problem for the Kuznetsov equation with initial-boundary conditions.
Let the function I 0 (t, y) = I 0 (t, √ εx ′ ) be L -periodic on t and such that for s ≥ n+1 2 and L 0 I 0 (s, y)ds = 0 . Hence [41], there is a unique solution I(τ, z, y) of the Cauchy problem (16) for the KZK equation satisfying (17). We define u and g as in Eqs. (20) and (22) respectively. Thus, for R Kuz−KZK defined in Eq. (15), u is the solution of the following system In the same time let us consider for a sufficiently large T > 0 the solution u (see Theorem 8 for its global existence and uniqueness) of the Dirichlet boundary-value problem for the Kuznetsov equation To compare u and u , we obtain the following stability result: Then, the following statements are valid.
1. If s ≥ 6 for n = 2 and 3 , or else s 2 > n 2 + 1 , there exists a constant C 0 > 0 such that I 0 H s < C 0 implies the global well-posedness of the Cauchy problem for the KZK equation with the following regularity: Moreover it implies the well-posedness of (27) with The imposed regularity of I 0 (see Table 1) is minimal to ensure that R Kuz−KZK (see Eq. (15) for the definition) is in C([0, +∞[; L 2 (R + × R n−1 )) .
2. If s 2 > n 2 + 2 , taking the same initial data for the exact boundary-value problem for the Kuznetsov equation (28) as for u , i.e.
there exists C 0 > 0 such that I 0 H s < C 0 implies the well-posedness of the exact Kuznetsov equation (28) supplemented with the Dirichlet boundary condition ensuring the regularity Moreover, there exist constants K , C , 3. In addition, let u be a solution of the Dirichlet boundary-value problem (28) for the Kuznetsov equation, with g defined by Eq. (22) and with initial data u 0 ∈ H m+2 (Ω) , Then there exist constants K , C , Proof : Let u and g be defined by Eqs. (20) and (22) respectively by the solution I of the Cauchy problem (16) for the KZK equation with I| z=0 = I 0 ∈ H s (T t × R n−1 ) and s ≥ 6 for n = 2 and 3 , or else s 2 > n 2 + 1 . In this case, u is the global solution of the approximated Kuznetsov system (27), what is a direct consequence of Theorem 1.2 in Ref. [41]. If I 0 ∈ H s (T t × R n−1 ) with the chosen s , then I ∈ C({z > 0}; H s (T τ ×R n−1 )) . But knowing, thanks to estimate (19), that ∆ k y I 0 ∈ H s−2k (T t ×R n−1 ) implies also ∂ −k t ∆ k y I 0 ∈ H s−2k (T t ×R n−1 ) for 1 ≤ k ≤ s 2 , the condition in [41, Thm. 1.2, Point 4] is verified and thus we have the following regularity of I on z : for 0 ≤ k ≤ s 2 I(τ, z, y) ∈ C k ({z > 0}; H s−2k (T τ × R n−1 )).
As u is defined by (20), we deduce (using as previously the notation Ω 1 = T τ × R n−1 ) u(τ, z, y) and ∂ τ u(τ, z, y) but we can also say [21,41], thanks to the exponential decay of the solution of the KZK equation on z , that u(τ, z, y) and ∂ τ u(τ, z, y) ∈H k ({z > 0}; H s−2k (Ω 1 )), ). This implies for the chosen s that . For the chosen s these regularities of u(t, x 1 , x ′ ) give us regularity (29) and allow to have all left-hand terms in the approximated Kuznetsov system (27) of the desired regularity, i.e C([0, +∞[; L 2 (Ω)) . In addition for s 2 > n 2 + 2 with the chosen g , u 0 = u(0) and u 1 = u t (0) in the conditions of the theorem we have Let us now prove (33) from point 3 as it directly implies estimate (31) from point 2 . We subtract the Kuznetsov equation from the approximated Kuznetsov equation (see system (27)), multiply by (u − u) t and integrate over Ω to obtain, as in Ref. [11], the following stability estimate:
Remark 2 Let us discuss the corresponding approximation results in the inviscid case. We have two approximation results: 1. between the solutions U KZK of the KZK equation and U Euler of the Euler system [12,42] (see [12,Thm 6.8] for the definitions of U Euler and U KZK ) in a cone

Consequently, we obtain the approximation result between the solutions U KZK of the KZK equation and the solutions U Kuzn of the Kuznetsov equation in C(T ) by the triangular inequality:
U Kuzn − U KZK 2 L 2 (Qε(t)) ≤ K(ε 3 t + δ 2 )e Kεt ≤ 9ε 2 , as soon as (U Kuzn − U KZK )(0) L 2 (Qε(0)) ≤ δ < ε . The initial data are constructed on the initial data I 0 for the KZK equation. More precisely we take I 0 ∈ H s (T τ × R n−1 ) for s > max{10, n 2 + 1}, which ensures in the case of the same initial data

Approximation of the solutions of the Kuznetsov equation with the solutions of the NPE equation.
Now let us go back to the NPE equation (11) and consider its ansatz (see [12] for the derivation of the NPE equation from the isentropic Navier-Stokes system or the Euler system). In contrast with Eq. (12) for the KZK equation, this time the velocity potential is given [43] by u(x, t) = Ψ(εt, x 1 − ct, √ ǫx ′ ) = Ψ(τ, z, y).
Thus we directly obtain from the Kuznetsov equation (9) with the paraxial change of variable that We obtain the NPE equation satisfying by ∂ z Ψ modulo a multiplicative constant: In the sequel we work with ξ defined by which solves the Cauchy problem for the NPE equation in the class of L− periodic functions with respect to the variable z and with mean value zero along z . The introduction of the operator ∂ −1 z defined similarly to ∂ −1 τ in Eq. (18) allows us to consider instead of Eq. (11) the following equivalent equation This time, in comparison with the KZK equation, we use the bijection between this two models (see [12]). We also update our notation for Ω 1 = T z × R n−1 y and take s > n 2 + 1 . Suppose that ξ 0 ∈ H s+2 (T z × R n−1 y ) and Tz ξ 0 (z, y) dz = 0.
We define ∂ x 1 u(t, x 1 , x ′ ) := − c ρ 0 ξ(τ, z, y) with the change of variable (35) and Thus for these initial data there exists with R Kuz−N P E defined in Eq. (36). If we consider the Cauchy problem for the Kuznetsov equation on T x 1 × R n−1 x ′ with u 0 and u 1 derived from ξ 0 , we have Hence, if ξ 0 H s+2 (Tz ×R n−1 y ) is small enough [11], we have a unique bounded in time solu- of the Cauchy problem for the Kuznetsov equation (40).
Theorem 3 For ν ≥ 0 let u and u be the defined above solutions of the exact Cauchy problem (40) and of the approximated Cauchy problem (39) for the Kuznetsov equation on Ω = T x 1 × R n−1 x ′ respectively. Then for ν > 0 there exist K , C , C 1 , C 2 > 0 such that for all t < C ε estimate (31) is valid and in addition it holds Point 3 of Theorem 2. Moreover, if for n ≤ 3 , and ξ 0 ∈ H s (T x 1 × R n−1 ) with s ≥ 4 , then the approximated solution satisfies

then the approximated solution satisfies
). Under these conditions for n ≥ 1 We can see for n = 2 or 3 , using the previous arguments that the minimum regularity of the initial data (see Table 1) to have the remainder terms ). In the same way for n ≥ 4 we find the minimal regularity for ξ 0 ∈ H s (Ω) with s > n 2 + 2 as it implies ). The optimality of the previously chosen s also comes from the fact that in Eq. (36) the least regular term in R Kuz−N P E is ∂ τ Ψ∂ 2 τ Ψ presenting for both viscous and inviscid cases.

Derivation of the Westervelt equation from the Kuznetsov equation.
Let u be a solution of the Kuznetsov equation (9). Similarly as in Ref. [1] we set and obtain By definition (41) of Π we have where We recognize the Westervelt equation (8) obtained up to remainder terms of order ε 2 .

Approximation of the solutions of the Kuznetsov equation by the solutions of the Westervelt equation
The well-posedness of the Westervelt equation follows directly from [11]. For a solution of the Cauchy problem (40) for the Kuznetsov equation u we define as in Subsection 4.1 Π by Eq. (41). Hence Π is the solution of the approximated Cauchy problem for the Westervelt equation (42) with the initial data defined by with u 0 and u 1 initial data of the Cauchy problem (40) for the Kuznetsov equation. For s > n 2 and ν > 0 , if we take u 0 ∈ H s+3 (R n ) and In the inviscid case when ν = 0 , for s > n 2 if we still take u 0 ∈ H s+3 (R n ) , but u 1 ∈ H s+2 (R 3 ) , we have Π 0 ∈ H s+2 (R n ) and Π 1 ∈ H s+1 (R n ) with the estimate Then, similarly to our previous work [11], we obtain the following result.
Theorem 4 Let n ≥ 1 and s > n 2 . 1. If ν > 0 , u 0 ∈ H s+3 (R n ) and u 1 ∈ H s+3 (R n ) , then there exists a constant k 2 > 0 such that for the exact Cauchy problem for the Westervelt equation with Π 0 and Π 1 defined by Eqs. (45) and (46), has a unique global in time solution Moreover, Π , obtained from the solution u of the Kuznetsov equation with Eq. (41), is the unique global in time solution of the approximated Cauchy problem (42), (44) with the same regularity as Π .
Lemma 1 Let n ∈ N * and s ≥ 0 . There exists a unique solution of the periodic in time boundary value problem for the linear strongly damped wave equation (56) u ∈ X = u ∈ H 2 (T t ; H s (Ω)) ∩ H 1 (T t ; H s+2 (Ω))| Tt u(s, x) ds = 0 ∀x ∈ Ω (59) if and only if f and g satisfy along with (58). Moreover the following stability estimate holds .
Here H 2 (T t ; H s (Ω)) ∩ H 1 (T t ; H s+2 (Ω)) is endowed with its usual norm denoted here and in the sequel by . X .
To prove the global well-posedness of the periodic in time boundary value problem (21) for the Kuznetsov equation we use its boundary condition as the initial condition of the corresponding Cauchy problem in R n and we combine the maximal regularity result for system (56) with [48, 1.5 Cor., p. 368] (see also [11,Thm. 4.2]) applying the same method as previously done for the Cauchy problem associated with the Kuznetsov equation [11].
Theorem 7 Let ν > 0 , n ∈ N * and s > n 2 . Let X be defined by ( 59 ) and the boundary condition g ∈ F T be defined by (60) and in addition, let g be of mean value zero (see Eq. (58)).
Then there exist r * = O(1) and C 1 = O(1) such that for all r ∈ [0, r * [ , if g F T ≤ √ νε C 1 r, there exists a unique solution u ∈ X of the periodic problem (21) for the Kuznetsov equation such that u X ≤ 2r .
Proof : For g ∈ F T defined in (60) and satisfying (58), let us denote by u * ∈ X the unique solution of the linear problem (56) with f = 0 and g ∈ F T . In addition, according to Theorem 1, we take X defined in (59), this time for s > n 2 (we need this regularity to control the non-linear terms), and introduce the Banach spaces and Then by Lemma 1, the linear operator is a bi-continuous isomorphism.
Let us now notice that if v is the unique solution of the non-linear Dirichlet problem then u = v + u * is the unique solution of the periodic problem (21). Let us prove the existence of a such v , using [48, 1.5 Cor., p. 368].
We suppose that u * X ≤ r and define for v ∈ X 0 For w and z in X 0 such that w X ≤ r and z X ≤ r , we estimate the norm Φ(w) − Φ(z) Y . By applying the triangular inequality we have Now, for all a and b in X with s ≥ s 0 > n 2 it holds where C H 1 (Tt;H s 0 (Ω))→L ∞ (Tt×Ω) is the embedding constant of H 1 (T t ; H s 0 (Ω)) in L ∞ (T t × Ω) , independent of s , but depending only on the dimension n . In the same way, for all a and b in X it holds Taking a and b equal to u * , w , z or w − z , as u * X ≤ r , w X ≤ r and z X ≤ r , we obtain By the fact that L is a bi-continuous isomorphism, there exists a minimal constant C ε = O 1 εν > 0 , coming from the inequality for u , a solution of the linear problem (56) with homogeneous boundary data (for a maximal constant C 0 = O(1) > 0 ) such that Then we find for w and z in X 0 , such that w X ≤ r , z X ≤ r , and also for u * X ≤ r , that with the notation Thus we apply [48, 1.5 Cor., p. 368] with f (x) = L(x) − Φ(x) and x 0 = 0 . Therefore, knowing that C ε = C 0 εν , we have, that for all r ∈ [0, r * [ with there exists a unique v ∈ 0 + rU X 0 such that L(v) − Φ(v) = y . Since we are seeking v , which solves the non-linear problem (62), we need to impose y = 0 , i.e. that v be the solution of the non-linear problem (62), then we need to impose y = 0 and thus, to ensure that 0 ∈ Φ(0) + w(r)LU X 0 .
Since − 1 w(r) Φ(0) is an element of Y and LX 0 = Y , there exists a unique z ∈ X 0 such that Let us show that z X ≤ 1 , what will implies that 0 ∈ Φ(0) + w(r)LU X 0 . Noticing that and using (64) we find as soon as r < r * . Consequently, z ∈ U X 0 and Φ(0) + w(r)Lz = 0 . Then we conclude that for all r ∈ [0, r * [ , if u * X ≤ r , there exists a unique v ∈ rU X 0 such that L(v) − Φ(v) = 0 , i.e. v is the solution of the non-linear problem (62). Thanks to the maximal regularity and a priori estimate following from Theorem 1 with f = 0 , there exists a constant Thus, for all r ∈ [0, r * [ and g F T ≤ √ νε C 1 r , the function u = u * + v ∈ X is the unique solution of the time periodic problem for the Kuznetsov equation and u X ≤ 2r .

A.2 Initial boundary value problem.
We still work on Ω = R + × R n−1 and we study the initial boundary value problem for the Kuznetsov equation on this space, i.e. the perturbation of an imposed initial condition by a source on the boundary, which in Subsection 2.2.2 was determined by the solution of the KZK equation.
Lemma 2 Let s ≥ 0 , n ∈ N . There exists a unique solution of the linear problem if and only if the data satisfy the following conditions • f ∈ L 2 (R + ; H s (Ω)), • for the boundary condition • u 0 ∈ H s+2 (Ω) and u 1 ∈ H s+1 (Ω) , • g(0) = u 0 and g t (0) = u 1 on ∂Ω in the trace sense.
In addition, the solution satisfies the stability estimate In order to prove this result we will use the following lemma to remove the inhomogeneity g .
To obtain uniqueness, let w be a solution to (68) with g = 0 . Since w t solves the heat problem with homogeneous data, we obtain w t = 0 and therefore also w = 0 by the initial condition w(0) = 0 . The stability estimate follows from the closed graph theorem.
Let us prove Lemma 2: Proof : We obtain the uniqueness of the solution of the boundary value problem for the linear strongly damped equation (66) from the fact that in the case g = 0 we can consider −∆ as a self-adjoint and non negative operator with homogeneous Dirichlet boundary conditions and we can use [15].
To verify the necessity of the conditions on the data, we suppose that u ∈ E (see Eq. (65) for the definition of E ) is a solution of (66). Then u, u t ∈ H 1 (R + ; H s (Ω)) ∩ L 2 (R + ; H s+2 (Ω)) and thus f ∈ L 2 (R + ; H s (Ω)).
Let u ∈ E , denote the restriction ofũ to Ω and let g := g − u| ∂Ω . By the spatial trace theorem u| ∂Ω ∈ F R + , and hence g ∈ F R + . Then the solution u of the non homogeneous linear problem (66) is given by u = v + u , where v solves problem (66) with f = u 0 = u 1 = 0 and g = g . >From Lemma 3 we have a unique solution v ∈ E u of problem (68) with g = g . Then the function w := v − v solves the following system w(0) = 0, w t (0) = 0 in Ω, which thanks to [15,Thm. 2.6] has a unique solution w ∈ E defined in (65). The function u := w + v + u is the desired solution of system (66) and the stability estimate follows from the closed graph theorem. This concludes the proof of Lemma 2.
The next theorem follows from the maximal regularity result of Lemma 2 and of [48, 1.5. Cor., p. 368]. Its proof is similar to the proof of Theorem 7 and hence is omitted.
Theorem 8 Let ν > 0 , n ∈ N * , Ω = R + × R n−1 and s > n 2 . Considering the initial boundary value problem for the Kuznetsov equation in the half space with the Dirichlet boundary condition (28) the following results hold: there exist constants r * = O(1) and C 1 = O(1) , such that for all initial data satisfying