Cauchy Problem for the Kuznetsov Equation

We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the non-viscous case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2-stability estimate, obtained between a regular and a less regular solutions.


Introduction
The Kuznetsov equation [16] models a propagation of non-linear acoustic waves in thermoviscous elastic media. This equation describes the evolution of the velocity potential and can be derived, as in [20], from a compressible isentropic Navier-Stokes system, for the viscous case, or the Euler system for the inviscid case, using small perturbations of the density and of the velocity characterized by a small dimensionless parameter ε > 0 . The Cauchy problem for the Kuznetsov equation reads for α = γ−1 c 2 , β = 2 and ν = δ ρ 0 as where c , ρ 0 , γ , δ are the velocity of the sound, the density, the ratio of the specific heats and the viscosity of the medium respectively. In what follows, we can just suppose that α and β are some positive constants. Eq. (1) is a weakly quasi-linear damped wave equation, that describes a propagation of a high amplitude wave in fluids. The Kuznetsov equation is one of the models derived from the Navier-Stokes system, and it is well suited for the plane, cylindrical and spherical waves in a fluid [7]. Most of the works on the Kuznetsov equation (1) are treated in the one space dimension [11] or in a bounded spatial domain of R n [12,13,17]. For the viscous case Kaltenbacher and Lasiecka [13] have considered the Dirichlet boundary valued problem and proved for sufficiently small initial data the global well-posedness for n ≤ 3 . Meyer and Wilke [17] have proved it for all n . In [12] it was proven a local well-posedness of the Neumann boundary valued problem for n ≤ 3 .
In this article we study the well-posedness properties of the Cauchy problem (1)- (2). In the inviscid case for ν = 0 , the Cauchy problem for the Kuznetsov equation is a particular case of a general quasi-linear hyperbolic system of the second order considered by Hughes, Kato and Marsden [8] (see Theorem 1 Points 1 and 2 for the application of their results to the Kuznetsov equation). The local well-posedness result, proved in [8], does not use a priori estimate techniques and is based on the semi-group theory. Hence, thanks to [8], we have the well-posedness of (1)- (2) in the Sobolev spaces H s with a real s > n 2 + 1 . Therefore, actually, to extend the local well-posedness to a global one (for n ≥ 4 ) and to estimate the maximal time interval on which there exists a regular solution, John [10] has developed a priori estimates for the Cauchy problem for a general quasi-linear wave equation. This time, due to the non-linearities u t u tt and ∇u ∇u t including the time derivatives, to have an a priori estimate for the Kuznetsov equation we need to work with Sobolev spaces with a natural s , thus denoted in what follows by m . If we directly apply general results of Ref. [10] to our case of the Kuznetsov equation, we obtain a wellposedness result with a high regularity of the initial data. We improve it in Theorem 3 and show John's results for the Kuznetsov equation with the minimal regularity on the initial data corresponding to the regularity obtained by Hughes, Kato and Marsden [8]. For instance, we prove the analogous energy estimates in H m with m ≥ [ n 2 + 2] instead of John's m ≥ 3 2 n + 4 (see Eq. (20) in Proposition 1) and its slight modified version in H m with m ≥ [ n 2 + 3] instead of m ≥ 3 2 n + 7 (see Eq. (21) in Proposition 2). The energy estimates allow us to evaluate the maximal existence time interval (see Theorem 1 Point 5 and Theorem 4 for more details). In R 2 and R 3 the optimality of obtained estimations for the maximal existence time is ensured by the results of Alinhac [2]. In Ref. [2] a geometric blow-up for small data is proved for ∂ 2 t u and ∆u at a finite time of the same order as predicted by our a priori estimates (see Theorem 1 Point 5, our estimates of the minimum existence time correspond to Alinhac's maximum existence time results). From the other hand, the blow-up of ∂ 2 t u and ∆u is also confirmed by the stability estimate (8) in Theorem 1: if the maximal existence time interval is finite and limited by T * , by Eq. (8), we have the divergence For n ≥ 4 and ν = 0 , we also improve the results of John [10] and show the global existence for sufficiently small initial data u 0 ∈ H m+1 (R n ) and u 1 ∈ H m (R n ) with m ≥ n + 2 instead of m ≥ 3 2 n + 7 (see Proposition 4 and Theorem 4). The smallness of the initial data here directly ensures the hyperbolicity of the Kuznetsov equation for all time, i.e. it ensures that 1 − αεu t is strictly positive and bounded for all time. The proof uses the generalized derivatives for the wave type equations [10] and a priori estimate of Klainerman [14,15] (see Section 3.2).
Let us now formulate our main well-posedness result for the inviscid case: Theorem 1 (Inviscid case) Let ν = 0 , n ∈ N * and s > n 2 +1 . For all u 0 ∈ H s+1 (R n ) and u 1 ∈ H s (R n ) such that u 1 L ∞ (R n ) < 1 2αε , u 0 L ∞ (R n ) < M 1 , ∇u 0 L ∞ (R n ) < M 2 , with M 1 and M 2 in R * + the following results hold: 1. For all T > 0 , there exists T ′ > 0 , T ′ ≤ T , such that there exists a unique solution u of (1)-(2) with the following regularity 3. Let T * be the largest time on which such a solution is defined, and in addition there exist constants C(n, c, α) > 0 andĈ(n, c, α, β) > 0 (see Theorem 3) such that if the initial data satisfies , such that it holds (3).
4. For two solutions u and v of the Kuznetsov equation for ν = 0 defined on [0, T * [ assume that u be regular as in (4) and with a bounded ∇v t L ∞ (R n ) norm on [0, T * [ . Then it holds the following stability uniqueness result: there exist constants C 1 > 0 and C 2 > 0 , independent on time, such that 5. If s = m ≥ n + 2 , then for sufficiently small initial data (see Theorem 4 in Section 3.2) (a) lim inf ε→0 ε 2 T * > 0 for n = 2, Theorem 1 is principally based on the a priori estimates given in Sections 3.1 (for Point 3) and 3.2 (for Point 5) and on the local existence result updated from Ref. [8] (Points 1 and 2). Point 4 uses the classical ideas of the weak-strong stability, for instance proved in details for the KZK equation in [18] Theorem 1.1 Point 4 p. 785. Hence its proof is omitted. Some technical details on the proof of the a priori estimates of Section 3.1 can be found in Appendix A.
Analyzing the structure of the Kuznetsov equation and the difficulties involving by its non-linear terms, we start in Section 2 with preliminary remarks on the L 2 -energy properties for the Kuznetsov equation to compare with its simplified versions. Developing the energy estimates in the Sobolev spaces, we however recognize the structure of the L 2 -energy of the wave equation which keeps unchanged.
In the presence of the term ∆u t for the viscous case ν > 0 , the regularity of the higher order time derivatives of u is different (to compare to the inviscid case), and the way to control the non-linearities in the a priori estimates becomes different. As it was shown in [21], this dissipative term changes a finite speed of propagation of the wave equation to the infinite one. Indeed, the linear part of Eq. (1) can be viewed as two compositions of the heat operator ∂ t − ∆ in the following way: For the viscous case we prove the global in time well-posedness results in R n (see Section 4) for small enough initial data, the size of which we specify (see Point 1 of Theorem 2 and Subsection 4.1 for its proof). In Subsection 4.2 for n ≥ 3 (see Point 2 of Theorem 2) we establish an a priori estimate which gives also a sufficient condition of the existence of a global solution for a sufficiently small initial energy of the same order on ε as in Point 1 of Theorem 2. The same results (see Point 3 of Theorem 2) hold in (R/LZ) × R n−1 for n ≥ 2 (with a periodicity and mean value zero on one variable).
Theorem 2 (Viscous case) Let ν > 0 , n ∈ N * , s > n 2 and R + = [0, +∞[ . Considering the Cauchy problem for the Kuznetsov equation (1)-(2), the following results hold: the initial data u 0 ∈ H s+2 (R n ) and u 1 ∈ H s+1 (R n ), r * = O(1) be the positive constant defined in Eq. (38) and C 1 = O(1) be the minimal constant such that the solution u * of the corresponding linear Cauchy problem (35) satisfies 3. For n ∈ N * in Ω = (R/LZ) × R n−1 with s = m ∈ N even and m ≥ [ n 2 + 3] there hold Points 1 and 2 in the class of periodic in one direction functions with the mean value zero Let us notice that the hyperbolicity condition (5) is also satisfied if we require conditions (9) and (11). For ν > 0 Point 4 of Theorem 1 obviously holds for all n ∈ N * . Point 1 of Theorem 2 is proved in Subsection 4.1 using a theorem of a non-linear analysis [22] (see Theorem 6) and regularity results for the strongly damped wave equation following [6], which can also be used for Ω = (R/LZ) × R n−1 in point 3. Point 2 of Theorem 2 is proved in Subsection 4.2, using a priori estimates given in Proposition 1, see also Theorem 7. The last point of Theorem 2 is a direct corollary of the Poincaré inequality which holds in the class of periodic functions with the mean value zero. Estimate (13) allows to have the same estimate as in Lemma 1 (see Section 4) for n = 2 , which fails in R 2 . Thus, it also gives the global existence for rather small initial data detailed in Point 2.

Preliminary remarks on L -energies
We can notice that Eq. (1) is a wave equation containing a dissipative term ∆u t and two non-linear terms: ∇u∇u t describing local non-linear effects and u t u tt describing global or cumulative effects. Actually, the linear wave equation appears from Eq. (1) if we consider only the terms of the zero order on ε : The semi-group theory permits in the usual way to show that for u 0 ∈ H 1 (R n ) and u 1 ∈ L 2 (R n ) there exists a unique solution of the Cauchy problem (14), (2) u ∈ C 0 (R + ; H 1 (R n )) ∩ C 1 (R + ; L 2 (R n )).
So the energy of the wave equation (14) is well defined and conserved d dt For ν > 0 and without non-linear terms, the Kuznetsov equation (1) becomes the known strongly damped wave equation: which is well-posed [9]: for m ∈ N , u 0 ∈ H m+1 (R n ) and u 1 ∈ H m (R n ) there exists a unique solution of the Cauchy problem (16), (2) u ∈ C 0 (R + ; H m+1 (R n )) ∩ C 1 (R + ; H m (R n )).
Multiplying Eq. (16) by u t in L 2 (R n ) , we obtain for the energy of the wave equation (15) d dt what means that the energy E(t) decreases in time, thanks to the viscosity term with ν > 0 . The decrease rate is found for more regular energies in [21] in accordance with the regularity of the initial conditions. Without the term ∇u∇u t (local non-linear effects), the Kuznetsov and can also be seen as an approximation of an isentropic Navier-Stokes system.
In the sequel we conveniently denote p by u . We multiply Eq. (17) by u t and integrate over R n to obtain Then we have For α = 2 3 γ+1 c 2 we consider the energy which is monotonous decreasing for ν > 0 and is conserved for ν = 0 . Let us also notice that, taking the same initial data for ν = 0 and ν > 0 , we have: in the assumption that 1 − αεu t ≥ 0 almost everywhere.
, that is to say u t (t) L ∞ (R n ) remains small enough in time, then we can compare E nonl to the energy of the wave equation Then a sufficiently regular solution of the Cauchy problem for the Westervelt equation has the energy E controlled by a decreasing in time function: Now, let us consider the Kuznetsov equation (1). We multiply it by u t and integrate on R n to obtain where E nonl (t) is given by Eq.
Thus, for α = 2 3 γ−1 c 2 , the function is constant if ν = 0 and decreases if ν > 0 . Let us notice that while 1 2 ≤ 1 − αεu t ≤ 3 2 , the coefficient c 2 − 2εu t is always positive (since c is the sound speed in the chosen medium, c 2 ≫ 1 ), hence the first integral in F ν (t) is positive, but we a priori don't know the sign of the second integral, i.e. the sign of u tt . However, for ν = 0 , F ν=0 (t) is positive, as soon as 0 ≤ 1 − αεu 1 : and, if we take the same initial data for the Cauchy problems with ν = 0 and ν > 0 , for For n ≥ 3 , we can control the term 2ε R n ∇u∇u t u t dx using the Hölder inequality and the Sobolev embeddings (which fails in R 2 ): Indeed, in R 2 we don't have any estimates of the form with p > 2 . But such an estimate is essential to control the nonlinear term. Then, instead of Eq. (19) for F ν , we have the relation for E nonl : So, if a solution of the Kuznetsov equation u is such that ∇u(t) L n and u t (t) L ∞ stay small enough for all time, then E nonl decreases in time and, as previously for the Westervelt equation, thanks to 1 2 E(t) ≤ E nonl (t) ≤ 3 2 E(t) , the energy E has for upper bound a decreasing function.
This fact leads us to look for global well-posedness results for the Cauchy problem for the Kuznetsov equation in the viscous case.
3 Well-posedness for the inviscid case

Proof of Point 3 of Theorem 1
Let us give an estimation of the maximum existence time for a solution of problem (1)-(2) with ν = 0 . For this we follow the work of John [10] with the use of a priori estimate. However we don't directly apply the general results of John, but we improve them for our specific problem as we can take less regular initial conditions in order to have suitable a priori estimates. (4) and (5) with constants B = (3+2c 2 ) min(1/2,c 2 ) > 0 , depending only on c , and C m > 0 , depending only on m , on the dimension n and on c (only if min(1/2, c 2 ) = c 2 ).
Proof : The proof is given in Appendix A.
Inequality (20), proved in Proposition 1, gives us an a priori estimate in order to have, with the help of the Gronwall Lemma, an estimation of the maximum existence time T * . However, when m increases, C m increases, and the maximum existence time, given by estimate (20), decreases whereas the initial conditions become more regular. Therefore, we prove the second a priori estimate (see Eq. (21)), playing a key role in order to avoid this problem: with a constant D m > 0 , depending only on m , on n and on c and the same constant B as in Proposition 1.
The proof of Eq. . and Here B and C m 0 are the constants from estimate (20) and C ∞ is the embedding constant from the embedding of the Sobolev space Proof : Thanks to Point 1 of Theorem 1, for u 0 ∈ H m+1 (R n ) , u 1 ∈ H m (R n ) and u 1 L ∞ (R n ) < 1 2αε there exists a unique solution u on an sufficiently small interval [0, T ] of problem (1)-(2) with ν = 0 , satisfying (4) and (5) without adding further conditions of regularity on u 0 and u 1 as it can be reduced on a smallness condition on u 0 H m+1 + u 1 H m . Let us take T 0 , as defined in Eq. (22), and show by induction (20) with m = m 0 . According to the Gronwall Lemma, applied to (20) where z(t) is the solution of the Cauchy problem for an ordinary differential equation This problem can be solved explicitly: We can see that, as long as 0 ≤ t ≤ T 0 , the function z(t) has the finite upper bound : Moreover, thanks to our notations, from where, using inequality (23), we find , Since Eq. (5) holds on all interval [0, T 0 ] , we can use the a priori estimate (21) and write that for all t ∈ [0, Applying the Gronwall Lemma, we obtain for t ∈ [0, Theorem 3 estimates the lifespan T * as at least of the order 1 ε , or more precisely, implies that lim inf This result is independent on the dimension n . However, much better estimations for the lifespan can be obtained, if we use an inequality that takes into account the time decay of the solutions for n > 1 , what we do in the next section. The generators of this group (the derivatives with respect to group parameters taken at the identity), here called generalized derivatives, include in addition to the derivatives ∂ t , ∂ x 1 , . . . , ∂ xn , first-order differential operators L α with α = 0, . . . , n and Ω ik with 1 ≤ i < k ≤ n : Definition 1 (Generalized derivatives [10]) The following operators are called the generalized derivatives. The operators (taken in this order) are denoted respectively by Γ 0 , . . . , Γ µ with µ = 1 2 (n 2 + 3n + 2) . For a multi-index A = (A 0 , . . . , A µ ) we write in the usual way Therefore, in the framework of the general derivatives, we define for m ∈ N Let us give a remarkable estimate proved in Ref. [15] by Klainerman: Thanks to Proposition 3, we improve the results of John [10] for the case of the Kuznetsov equation and state: Proposition 4 For n and m in N * , m ≥ n+ 2 , let u be a local solution on an interval [0, T ] of problem (1)-(2) with ν = 0 , satisfying (4) and (5) where µ is defined in Definition 1, C j and E ij depend only on |A| ≤ m , and A j1 and A j2 are multi-indexes, such that By hypothesis on u , and then, by integrating of Eq.
By summing for |A| ≤ m , we obtain Now we use the Klainerman inequality (26), noticing that, if we take m ≥ n + 2 , we have This finishes the proof. We use the a priori estimate (27) to improve our estimation of the lifespan T * as a function of n .
as long as and, for a small enough ε , T * = +∞ for n ≥ 4 , i.e. the solution u is global.
Proof : This is a direct consequence of the Gronwall lemma, used with the a priori estimate (27), as it is done by John in [10].

Remark 1
The estimations, given for T * in the case n = 1, 2, 3 , are optimal, as soon as, thanks to Alinhac [2], they give the existence time of a smooth solution of the same order as Alinhac's blow-up time, i.e. up to the time of a geometrical blow-up formation. Theorem 5 Let s ≥ 0 and X be the space defined in Point 1 of Theorem 2. Then the system . Moreover it holds the following a priori estimate with u X := u H 2 (R + ;H s ) + u L 2 (R + ;H s+2 ) + u t L 2 (R + ;H s+2 ) .
Proof : First we take f ∈ L 2 (R + ; H s (R n )), u 0 ∈ H s+2 (R n ) and u 1 ∈ H s+1 (R n ) . We use the ideas of [6] (see Eq. (4.26)). For the sake of clarity, let us take s = 0 . We take the inner product in L 2 (R n ) of the equation with −∆u t and integrate by parts: Using Young's inequality and integrating over [0, t] , we find Since the domain of −∆ is H 2 , we obtain that u, u t ∈ L 2 (R + ; H 2 (R n )), and u tt ∈ L 2 (R + × R n ), and hence, u ∈ X for s = 0 . For s > 0 , as the equation is linear, we perform the same proof, using the fact that, the operator Λ = (1 − ∆) 1 2 , defined by its Fourier transform by the formula (Λu)(ζ) = (1 + |ζ| 2 ) 1 2û (ζ), relies the norm of H s with the L 2 -norm: The uniqueness of u follows from the linearity of the operator and the uniqueness of the solution of system (29) in the case f = 0 [9]. Conversely, if u ∈ X solution of system (29), this implies that Thanks to Theorem III.4.10.2 in [3], it follows that u t ∈ C(R + ; H s+1 (R n )) . Then we have u(0) ∈ H s+2 (R n ) and u t (0) ∈ H s+1 (R n ) . Moreover, it reads directly from the definition of X , that f ∈ L 2 (R + ; H s (R n )) for u ∈ X . The a priori estimate follows from the closed graph theorem. Let us notice that Theorem 5 states that problem (29) has L 2 -maximal regularity (see [5] Definition 2.1) on R + .
To be able to give a sharp estimate of the smallness of the initial data and in the same time to estimate the bound of the corresponding solution of the Kuznetsov equation (see Point 1 of Theorem 2), we use the following theorem from [22], which allows us to establish our main result of the global well-posedness of the Cauchy problem for the Kuznetsov equation: Theorem 6 (Sukhinin) Let X be a Banach space, let Y be a separable topological vector space, let L : X → Y be a linear continuous operator, let U be the open unit ball in X , let P LU : LX → [0, ∞[ be the Minkowski functional of the set LU , and let Φ : X → LX be a mapping satisfying the condition Then for any r ∈ [0, r * [ and y ∈ f (x 0 ) + w(r)LU , there exists an x ∈ x 0 + rU such that f (x) = y .
Remark 2 If either L is injective or KerL has a topological complement E in X such that L(E ∩ U) = LU , then the assertion of Theorem 6 follows from the contraction mapping principle [22]. In particular, if L is injective, then the solution is unique. Now, we have all elements to prove Point 1 of Theorem 2: for all r ∈ [0, r * [ with r * = O(ε 0 ) = O(1) (to be defined), as soon as the initial data are small as then the unique solution u ∈ X satisfies u X ≤ 2r ( r = O(1) ).
Remark 3 It is very important to notice that here all physical coefficients of the Cauchy problem for the Kuznetsov equation are expressed to compare to the powers of ε ( ε is the dimensionless parameter caracterising the medium perturbation as explained in [19] and [20]). In particular, if we take into account in Point 3 of Theorem 1 that c 2 = O( 1 ε ) , we obtain the same types of smallness of the initial energy for the inviscid case as in Point 2 of Theorem 2: . But, if we want to understand the smallness of the initial data by their norms without the calculus of the initial energy, the results of Point 1 of Theorem 2 can be useful. The sharp character of Point 1 of Theorem 2 can be illustrated by the following direct energy estimation approach, presented in Appendix B.
Let suppose that Point 2 of Theorem 2 holds (see also Eq. (10)). Thus, for n ≥ 3 , then it follows in a sufficient way (see Appendix B for more details) that for u 0 ∈ H m+1 (R n ) and for u 1 ∈ H m (R n ) it holds which implies the existence of a unique global solution u ∈ C 0 (R + ; H m+1 (R n ))∩C 1 (R + ; H m (R n )) of problem (1) Thus we see that by this approach the sufficient condition to have for all t ≥ 0 E m 2 [u](t) bounded by a constant of order zero on ε is given by Eq. (34) and depends on the smooth properties of the initial data (more they are regular, more they should be small). Hence, it is much more restrictive to compare to (33).

Proof :
For u 0 ∈ H s+2 (R n ) and u 1 ∈ H s+1 (R n ) let us denote by u * ∈ X the unique solution of the linear problem In addition, according to Theorem 5, we take this time for s > n 2 (we need it to control the non-linear terms), and introduce the Banach spaces and Y = L 2 (R + ; H s (R n )) . Then by Theorem 5, the linear operator is a bi-continuous isomorphism.
Let us now notice that if v is the unique solution of the non-linear Cauchy problem then u = v + u * is the unique solution of the Cauchy problem for the Kuznetsov equation (1)-(2). Let us prove the existence of a such v , using Theorem 6. 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 by applying the triangular inequality Now, for all a and b in X with s ≥ s 0 > n 2 it holds where C H 1 (R + ;H s 0 )→L ∞ (R + ×R n ) is the embedding constant of H 1 (R + ; H s 0 ) into the space L ∞ (R + × R n ) , independent on 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 C 0 εν u 2 X ≤ f Y u X for u , a solution of the linear problem (29) with homogeneous initial data [for a constant Then we find for w and z in X 0 , such that w X ≤ r , z X ≤ r , and also with u * X ≤ r , that εr . Thus we apply Theorem 6 for 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 for all y ∈ Φ(0) + w(r)LU X 0 ⊂ Y with there exists a unique v ∈ 0 + rU X 0 such that L(v) − Φ(v) = y . But, if we want that v be the solution of the non-linear Cauchy problem (37), 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 (39), 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. the solution of the non-linear Cauchy problem (37). Thanks to the maximal regularity and a priori estimate following from inequality (31) with f = 0 , there exists a constant Thus, for all r ∈ [0, r * [ and u 0 H s+2 (R n ) + u 1 H s+1 (R n ) ≤ √ νε C 1 r , the function u = u * + v ∈ X is the unique solution of the Cauchy problem for the Kuznetsov equation and u X ≤ 2r .

Proof of Point 2 of Theorem 2: Case n ≥ 3
Knowing the existence of a solution u of the Kuznetsov equation in we notice that this directly implies that u ∈ C(R + ; H s+2 (R n )) and u t ∈ H 1 (R + ; H s (R n )) ∩ L 2 (R + ; H s+2 (R n )).
By Theorem III.4.10.2 in [3], it implies that u t ∈ C(R + ; H s+1 (R n )) , which gives that u ∈ C 1 (R + ; H s+1 (R n )) ∩ C(R + ; H s+2 (R n )) and, this time with the help of the Kuznetsov equation, u tt ∈ C(R + ; H s−1 (R n )) . Consequently, in the viscous case the regularity of the time derivatives of the order greater than two of the solutions differs from the regularity, obtained in Section 3 for the inviscid case. Thus we have to consider estimates with different energies: the energy E m 2 [u](t) , defined in Eq. (10), and the energy defined, as E m 2 [u](t) , for m ∈ N and m even, which respect to the obtained regularity of u and its derivatives. Lemma 1 Let n ∈ N * , n ≥ 3 , m ∈ N , and u be the solution of problem (1)- (2). Then for m ≥ n 2 + 3 , m even, and all multi- with a constant C m > 0 , depending only on m and on the dimension n .
Proof : Following notations of the proof of Proposition 1 in Annexe A, we redefine where u is the solution of problem (1). For this new L u v with the additional term νε∆v t , we have a modified version of relation (65) For n ≥ 3 , m ≥ n 2 + 3 and m even, we have, thanks to the Hölder inequality, .
Noticing, that, thanks to Ref. [ is the Banach space of bounded continuous functions equal to zero at the infinity), we can write for m ≥ n 2 + 3 In addition, with the help of the Gagliardo-Nirenberg-Sobolev inequality we also have

Therefore, all norms ∇D
, for the chosen n, m and A 0 , are present in S m 2 . Hence, we find and in the same way, . To calculate L u D A u we use expression (69) with multi-indexes A j1 and A j2 satisfying (70). As in the proof of Proposition 1, without loss of generality, we consider two multi-indexes A 1 and A 2 with the same properties (70). We perform two steps: Step 1 we prove Step 2 we prove Step 1. Thanks to properties (70) of A 1 and A 2 and to the symmetry of the general case we divide our proof on three typical cases: we consider the integrals of the form R n |(D we consider only non-trivial time derivatives R n |(D Step 1, Case 1. By the generalized Hölder inequality with 1 p + 1 q = n+2 2n , we have . By the Sobolev embeddings (72) of H m 1 ⊂ L p and H m 2 ⊂ L q with m 1 + m 2 = n 2 − 1 and 0 < m 1 < n 2 − 1 , we find where we have also applied the Gagliardo-Nirenberg-Sobolev inequality (45). Hence, Now we are looking for 0 < m 1 < n 2 − 1 , such that in order to have Since m 2 = n 2 − 1 − m 1 , and by (70), The last system, thanks to |A| + A 0 ≤ m , corresponding to the assumptions of the Proposition, is satisfied if n 2 ≤ m 1 + |A 1 | + A 1 0 ≤ m. Using (70), we find that Therefore, since for Case 1 |A 2 | ≥ 2 and A 2 0 ≥ 1 , recalling that (again by (70)) |A|+A 0 ≤ m , we obtain 1 ≤ |A 1 | + A 1 0 ≤ m − 1. Thus, we distinguish three sub-cases: For n ≥ 3 , |A 1 | + A 1 0 = 1 instead of finding m 1 , we notice, that we have only two possibility: either D A 1 = ∂ t and A 2 = A , which gives estimate (46), or For the last case, by the generalized Hölder inequality, we have For m ≥ n 2 + 3 the first norm in Eq. (52) can be estimated using the continuous embedding H s (R n ) ⊂ L n (R n ) holding for s > n 2 : . With the help of the Gagliardo-Nirenberg-Sobolev inequality (45), we also estimate the second norm in (52) and for the last one we directly have . Thus we obtain as previously estimate (47) of Step 1.
This permits to conclude Case 1 of Step 1.
Step 1, Case 3. Let us notice that thanks to relations (70), from |A 1 | = A 1 0 and |A 2 | = A 2 0 it follows |A| = A 0 . We start as usual with the generalized Hölder inequality 2n . Then we apply the Gagliardo-Nirenberg-Sobolev inequality (45) and its more general version, which can be viewed as the embedding of the Sobolev space W 1 q * (R n ) in the Lebesgue space L q (R n ) with 1 q = 1 q * − 1 n and 1 ≤ q * < n : 2n . We notice that if we want to use the Sobolev embeddings (72) to L p and to L q * , it is only possible if 1 p and 1 q * are smaller then 1 2 , or equivalently, if 1 p + 1 q * = n+4 2n < 1 . Knowing that n+4 2n < 1 for n ≥ 5 , n+4 2n > 1 for n = 3 and n+4 2n = 1 for n = 4 , we treat separately two cases: n ≥ 5 and n = 3 or 4 . For n = 3 or 4 , we choose p = n 2 and q = 2n n−2 , implying q * = 2 . Thus, for n = 3 we use the continuous embedding 2 ) and for n = 4 we use Recalling that m is even, and, by our assumption Thus for n = 3 and n = 4 we find estimate (47). Now, for n ≥ 5 , when 1 p + 1 q * = n+4 2n < 1 , we have with m 1 + m 2 = n 2 − 2 and 0 < m 1 < n 2 − 2 by the Sobolev embeddings (72) which give us H m 1 ⊂ L p and H m 2 ⊂ L q * . We look for m 1 such that As m 2 = n 2 − 2 − m 1 and A 2 0 = A 0 + 1 − A 1 0 , system (57) is equivalent to As m − 2A 0 ≥ 0 , it is sufficient to have m 1 such that then necessary A 2 0 = A 0 , and using estimates (44) and (53) we directly find If A 1 0 = m 2 we are in a symmetric case as A 2 0 = 1 . This conclude the proof of Case 3 and of Step 1, i.e. of estimate (47).
Step 2. Let us show estimate (48). Thanks to properties (70) of A 1 and A 2 and to the symmetry of the general case we divide our proof on two typical cases: we consider the integrals of the form R n |(D with m 1 + m 2 = n 2 and 0 < m 1 < n 2 . Let us find m 1 with 0 < m 1 < n 2 such that in order to have As m 2 = n 2 − m 1 , |A 1 | + |A 2 | = |A| + 1 , and A 1 0 + A 2 0 = A 0 + 1 , system (58) is equivalent to m 1 + |A 1 | + A 1 0 ≤ m + 2, n 2 + |A| + A 0 + 2 ≤ m + 2 + m 1 + |A 1 | + A 1 0 . By our assumption |A| + A 0 ≤ m , and hence the last system is satisfied if m 1 verifies In our case A 1 0 > 0 , thus 2 ≤ |A 1 | + A 1 0 ≤ m , which implies the existence of a such . This concludes Case 1 of Step 2.
Case 2. Thanks to (70), the conditions |A 1 | > 0 with A 1 0 = 0 imply that |A| − A 0 > 0 . Consequently, with m 1 + m 2 = n 2 and 0 < m 1 < n 2 as in the previous case, we obtain In the aim to have , we need to find m 1 with 0 < m 1 < n 2 , such that By assumptions of this case it tolds 1 ≤ |A 1 | ≤ m , what guarantees the existence of such m 1 with 0 < m 1 < n 2 . Indeed, if 1 ≤ |A 1 | < n 2 , then we can take m 1 = n 2 − |A 1 | , and if n 2 ≤ |A 1 | ≤ m − 1 , then we can take m 1 = 1 2 . In the case |A 1 | = m , corresponding to D A 2 = ∂ t , we directly obtain This completes the proof of Step 2 and hence the proof of estimate (48). Thus, estimates (47) and (48) imply from where follows (43).
Thanks to Lemma 1, we have the following energy decreasing result: Theorem 7 Let n ≥ 3 , m ∈ N be even and m ≥ n 2 + 3 . For u 0 ∈ H m+1 (R n ) and u 1 ∈ H m (R n ) , satisfying the smallness condition according to Point 1 of Theorem 2, there exists a unique global solution of problem (1)-(2) and the energy E m 2 [u](0) < ∞ is well-defined. Then 1. it holds the a priori estimate where, denoting by V the set of all multi-indexes

if in addition
Proof : We sum (41) on all A ∈ V to obtain d dt and consequently, Thus, if for all time E(t) < √ 2ν max(α,β)Cm , and in particular, then we have the decreasing of E in time: Moreover, for all time t ≥ 0 .

A Proof of Proposition 1
Following [10], let us consider where u is a local solution on [0, T ] of problem (1)-(2) with ν = 0 , satisfying (4) and (5) for s = m . We multiply Eq. (62) by v t and integrate over R n Hence, denoting by we have the following equation Let A = (A 0 , A 1 , ..., A n ) be a multi-index, and For m ≥ n 2 + 2 and a multi-index A with |A| ≤ m we estimate, thanks to the definition of E m [u] by Eq. (6), with a constant C > 0 , depending only on n by the Sobolev embedding [1] Theorem 7.57 p. 228 In the same way, using the Sobolev embedding (67), we obtain To calculate L u D A u we apply the chain rule of differentiation to D A L u u = 0 . As where j is a finite sum, with C j and E ij depending only on |A| ≤ m , and A j1 and A j2 are multi-index such that Let us show for m ≥ n 2 + 2 the estimate Without loss of generality, we consider two multi-indexes A 1 and A 2 satisfying (70) and divide the proof of (71) in two parts: we estimate R n |D A 1 u t D A 2 u t D A u t |dx first, and As the proof of each part is very similar, we give the details only for the first one.
To estimate R n |D A 1 u t D A 2 u t D A u t |dx , we consider three cases: Case 2 |A 1 | ≤ m and |A 2 | = 1 , Case 3 |A 2 | ≤ m and |A 1 | = 1 .
Let us detail Case 1 (other cases can be treated in a similar way). For 2 ≤ |A 1 | ≤ m − 1 and 2 ≤ |A 2 | ≤ m − 1 , it holds with 1 p + 1 q = 1 2 by the general Hölder inequality [4]. Hence, using the Sobolev embedding [1] H m 1 (R n ) ֒→ L p (R n ) with 1 p = 1 2 − m 1 n and 0 < m 1 < n 2 , we find In what follows by C > 0 is denoted an arbitrary constant depending only on m and on n .
We have D A 1 u t H m 1 (R n ) ≤ ∂ . We need to find m for which there exists m 1 with 0 < m 1 < n 2 , such that or equivalently, by (70) |A 2 | = |A| + 1 − |A 1 | , if |A 1 | = 2 we can take m 1 = n 2 − 1 4 , if 2 < |A 1 | < n 2 + 1 we can take m 1 = n 2 + 1 − |A 1 | , if n 2 + 1 ≤ |A 1 | ≤ m − 1 we can take m 1 = 1 4 . Moreover, Then, thanks to relations (73), we conclude Consequently, for m ≥ n 2 + 2 , and A 1 and A 2 , satisfying properties (70), it holds By the same argument, for m ≥ n 2 + 2 and A 1 and A 2 , satisfying properties (70), we control the terms of the form R n |D A 1 ∂ x i u D A 2 ∂ x i u D A u t |dx : By the hypothesis that u is a local solution of the inviscid Kuznetsov equation, u satisfies Eq. (5), i.e. u t (t) L ∞ ≤ 1 2αε on [0, T ] , which implies the equivalence of energies We integrate relation (65) over [0, t] with t ≤ T to obtain Then, using estimate (76), we find As we have this for all multi-index A with |A| ≤ m , by summing, we obtain with a constant C > 0 , depending only of n and m . This gives estimate (20).

Remark 4
To prove estimate (21) it is sufficient to show, using the proof of Proposition 1, that for m ≥ n 2 + 3 and all multi-index A with |A| ≤ m with a 0 = 1 , a 1 = 2c 2 + 2 and a k+1 = a k + 2c 2 a k−1 + 2 k i=0 a i + 1 for 1 ≤ k ≤ m 2 − 1.
As for a small enough ε it holds 1 1−αεu 1 ∞ ≤ 2 , taking the . H m−2 (R n ) − norm of the last equality we obtain