Fujita–Kato solution for compressible Navier–Stokes equations with axisymmetric initial data and zero Mach number limit

In this paper, we investigate the question of the existence of global strong solution for the compressible Navier–Stokes equations for small initial data such that the rotational part of the velocity [Formula: see text] belongs to [Formula: see text] (in dimension [Formula: see text]). We show then an equivalent of the so-called Fujita–Kato theorem to the case of the compressible Navier–Stokes equations when we consider axisymmetric initial data. The main difficulty is linked to the fact that in this case the velocity is not Lipschitz, as a consequence we have to study carefully the coupling between the rotational and irrotational part of the velocity. In a second part, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero.


Introduction
The motion of a slighty compressible barotropic fluid is described by the following system: Here ρ = ρ (t, x) ∈ R and u = u (t, x) ∈ R N stand for the dimensionless density and velocity field, and the pressure P is a suitably smooth function of ρ . Unless otherwise specified, it will always be assumed that x belongs to the whole space R N (N ≥ 2). We denote by λ and µ the two Lamé coefficients of the fluid, which are constant and satisfy µ > 0 and ν = λ + 2µ > 0. Such a condition ensures ellipticity for the operator µ∆ + (λ + µ)∇div and is satisfied in the physical cases (where λ + 2µ N ≥ 0). The initial conditions (ρ 0 , u 0 ) are given. The parameter ε corresponds to the so called Mach number which is given by ε = LT −1 ξ −1 where L and T are the typical values of the length and time (before rescaling) and ξ stands for the sound speed. We refer to [20,25] or in the introduction of [30] for more explanations on the derivation of the above model. In this paper our purpose is double, indeed in a first time we are interested in proving the global existence of strong solution for small initial data for suitable initial data such that 1 the rotational part of the velocity verifies the same assumption as the so called Fujita Kato theorem and in fact the more general theorem due to Cannone-Meyer-Planchon (see [4]) . In a second time we will study the limit of such solution when the Mach number ε goes to zero, and we will recover as a limit a solution of the incompressible Navier Stokes equations. When we study equations derived from the fluids mechanics, a first interesting question is to find the largest functional space X in local or global well-posedness may be proved and after to compare this space X with the energy space. It is then natural to understand in which sense a space is critical for the existence of strong solution. That it is why it is important to study the scaling of the equations and the notion of self similar solution. In view of the celebrated works by H. Fujita and T. Kato for the incompressible model (N SI) (see [18]), one can guess that the critical space for the velocity isḢ 1 2 . By critical, we mean that we can solve the system (N SI) in functional spaces with norm invariant by the changes of scales which leave (N SI) invariant. Indeed we can check that for a solution v we can associate via the following transformation: (v 0 (x), f (t, x)) → (lv 0 (lx), l 3 f (l 2 t, lx)), v(t, x) → lv(l 2 t, lx) a family of solution to (N SI) and that the norm inḢ N 2 −1 is invariant by the transformation v 0 (x) → lv 0 (lx).In the case of barotropic fluids, we can observe that the transformations: (ρ 0 (x), u 0 (x), f (t, x)) → (ρ 0 (lx), lu 0 (lx), l 3 f (l 2 t, lx)), (ρ(t, x), u(t, x)) → (ρ(l 2 t, lx), lu(l 2 t, , lx)).
( 1.2) have that property, provided that the pressure term has been changed accordingly. Roughly speaking we expect that such spaces are optimal in term of regularity on the initial data in order to prove the well-posedness of the system (1.1). It seems then natural to consider initial data (b 0 , u 0 ) inḢ is not a subalgebra of L ∞ , in particular it does not allow to control the nonlinear term involving the density as the pressure since we can not use classical composition theorem. Furthermore if we wish to propagate any regularity on the density along the time, we have to observe that the density is governed by a transport equation. It requires then that the velocity should be Lipschitz if we want to apply classical theorem on the transport equation. These two restrictions led to consider slightly smaller space, the homogeneous which is also critical according (1.2). Indeed we can verify that the only critical Besov space B Danchin in [8] shows for the first time a result of existence of global strong solution with small initial data in critical space for the scaling of the system. More precisely the initial data are chosen as follows (q 0 , u 0 ) ∈ (B The main difficulty consists in obtaining suitable estimates on the linearized system with convection terms. The crucial point is the proof of damping effect on the density which enables to control the pressure term. This last result has been generalized to the case of Besov space constructed on L p space by Charve and Danchin in [5], Danchin in [12], Chen et al in [7] and the author in [21] by using two different methods. We refer also to [24] for the existence of global strong solution for the shallow water system with a smallness condition which is supercritical. In [6] we weaken the previous results in the case of the shallow water system by assuming that the initial velocity verifies only: Here we can observe that the choice of u 0 in B −1 ∞,1 is absolutely crucial since it allows to control the Lipschitz norm of the velocity u. This last result has been extended by Danchin and He in [13] to the case of the system (1.1) with constant viscosity coefficients and with the rotational part of the velocity Pu 0 which belongs only to B N 2 −1 2,r with r suitably chosen and such that r > 2.
We can mention that all these previous results require a Lipschitz control on the velocity and in particular that u 0 belongs at least to B −1 ∞,1 . We would like to avoid this restriction and to consider only the so called Fujita Kato initial data which corresponds to Pu 0 ∈Ḣ N 2 −1 . To do this, we are going to consider initial data with geometrical assumption in order to weaken the coupling in the mass equation between the density and the rotational part of the velocity Pu. More precisely we are going to consider axisymmetric initial data in dimension N = 2, 3. Let us give now more details on the form of the solution that we are going to consider. If we set ρ = 1 + εb , we are to study: In dimension N = 2 we are interested in looking at solutions of the form: We observe that the system (1.3) can be rewritten as follows: (1.5) We will also consider the case of axisymmetric solution when N = 3 which corresponds to rewrite (ρ, u) as follows: u ε (t, x) = u r,ε (t, r, z)e r + u θ,ε (t, r, z)e θ + u z,ε (t, r, z)e z = u r,ε + u θ,ε + u z,ε and ρ(t, x) = ρ ε (t, r, z), (1.6) with: x ⊥ |x | and e z = (0, 0, 1). As previously the system (1.3) can be rewritten as follows: ∂ t u r,ε + (u r,ε + u z,ε ) · ∇u r,ε − µ∆u r,ε + (λ + µ)[∂ r divu r,ε + ∂ r divu z,ε ]e r 1 + εb (1.7) Setting u k,ε = u r,ε + u z,ε we can rewrite the previous system as follows: ∂ t u k,ε + u k,ε · ∇u k,ε − µ∆u k,ε + (λ + µ)∇divu k,ε 1 + εb ∂ t u θ,ε + u k,ε · ∇u θ,ε − µ∆u θ,ε 1 + εb + u θ,ε · ∇u r,ε = 0. (1.8) We can observe that in the system (1.3) and (1.7) the coupling between the density b ε and the velocity u ε is weaken since only the irrotationnal part of the velocity governes the behavior of the density in the mass equation (except in the 3D case where Pu z,ε plays also a role). This remark will be crucial in order to choose u 2 0 or u θ,0 inḢ N 2 −1 . It will be however mandatory to choose u 1 0 , u r,0 and u z,0 in B in order to ensure a Lipschitz control on the irrotational part which allows us to estimate the density via the mass equation. An another difficulty will consists in dealing with the coupling between the rotational and the irrotational part of the velocity which appears via the terms u 2,ε · ∇u 2,ε and u θ,ε · ∇u θ,ε . We will show some regularizing effects induced on these bilinear convection terms which enables us to propagate the regularity B Let us focus now on the zero Mach number limit when ε goes to 0. A large amount of literature has been devoted to this main feature.Roughly speaking, two different heuristics have been developed. The case of well-prepared data which corresponds to the assumption that ρ 0 = 1 + O(ε 2 ) and divu 0 = O(ε) has been investigated in [27,28] and [25].
In this paper, we will consider the case of ill-prepared data, where it is only assumed that ρ 0 = 1 + εb ε 0 with (b 0 , u 0 , f ε ) uniformly bounded (in a convenient functional space), Pu 0 tending to some v 0 and Pf tending to some g when ε goes to 0. One expects that the limit v of u solves the the so called incompressible Navier-Stokes equations: Let us recall now some results of global strong solution for the system (N SI) which are due to Fujita and Kato in [18] and Cannone, Meyer and Planchon in [4] (see also [13]).
then (N SI) has a unique global solution u in the space: for M a constant depending only on N and p.
We can observe that under the condition (1.4) in dimension N = 2 and (1.6) in dimension N = 3 the previous system can be respectively rewritten as follows: and: (1.10) The main difficulty in order to justify rigorously the convergence is that one has to face the propagation of acoustic waves with the speed ε −1 , a phenomenon which does not occur in the case of "well-prepared" data. Nevertheless, several remarkable results have been obtained recently. First of all, for initial data with minimal regularity assumptions, P-L Lions stated in [31] the existence of global law considered is of type P (ρ) = aρ γ with certain restrictions on γ depending on the space dimension N . Since then, convergence results to the incompressible model have been proved by B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi. The case of periodic boundary conditions has been investigated in [32], the case of bounded domains with Dirichlet conditions in [16] and the case of the whole space in [15]. Some local weak convergence results are also available in a more general context (see [33]). Roughly, the main difference between the whole space and the periodic case is that in the former case one can utilize the dispersion of sound waves to get strong convergence results whereas in the latter case, the sound waves will oscillate forever, leading only to weak convergence.
In the framework of strong periodic solutions, several recent works have to be mentioned. For smooth initial data and no external force, it has been stated in [20] that slightly compressible two-dimensional periodic incompressible Navier-Stokes equations and is unlikely to extend in higher dimension. More recently Danchin was the first to justify rigorously the zero Mach limit in the framework of critical Besov space for the scaling of the equations (see [9,10]). More precisely for the hybrid Besov spaces). Danchin and He (see [13]) have recently generalized this result to the case of Besov spaces constructed on L p Lebesgue spaces with a Lipschitz control on the velocity. To finish we would like to mention a very interesting and original result due to Danchin and Mucha [14] who propose a new approach to pass from the compressible Navier-Stokes equation to the incompressible Navier-Stokes equation by considering the limit of the solution when λ the second coefficient of viscosity goes to +∞. In this paper we would like to extend the previous result [13] for initial data of the type Cannone-Meyer-Planchon see [4] for axisymmetric initial data. From now for the sake of simplicity, we shall also assume that the data (b 0 , u 0 ) do not depend on ε and will be merely denoted by (b 0 , u 0 , f ) (therefore v 0 = Pu 0 ). Our first main results concern the existence of global strong solution for small initial data when the Mach number is fixed ε = 1. We refer to the section 2 for the definition of the hybrid Besov space.
Let P be a suitably smooth function of the density. (1.12) If p = 2 the solution is unique.
Remark 1 To the best of our knowledge, it is the first result of global strong solution for the compressible Navier-Stokes equation with a velocity u which is not necessary Lipchitz.
In addition it allows to choose initial data only a the level of the energy concerning the rotational part. Let us mention that in [23] we prove the existence of global strong solution with large initial data for the system (1.1) in the case of viscosity coefficients of the type shallow water (µ(ρ) = µρ and λ(ρ) = 0). Theorem 1.3 Let ε = 1 and N = 3. Let P be a suitably smooth function of the den- (1.14) If p ≤ 3 the solution is unique.
Remark 2 This result is a equivalent to the so-called Fujita-Kato theorem and Cannone-Meyer-Planchon theorem (see [18,4]) for the compressible Navier-Stokes system in the case of axisymmetric initial data.
Definition 1.1 We refer to (2.21) for the definition of low and high frequencies for Besov space. We denote by E p ε,ν the space of functions (b, u) such that : . We endow that space with the following norm: We can now consider the global convergence of the solutions of (1.1) to (N SI) in the critical functional setting described above.
Theorem 1.4 Assume now that the initial data (b ε 0 , u ε 0 ) verify the regularity conditions of theorem 1.2 and 1.3. Let ε = εν. There exists a constant η independent of ε and ν such that if 1 : • When N = 2: then the following results hold: such that for some constant M > 0: Furthermore u 1 ε , u ε r , u ε z converge weakly to 0 and u 2 ε , u ε θ converges in the sense of distribution to v solution of (N SI).
Remark 3 It would be easy to generalize the previous result when N = 3 to the case For simplicity we just take r 1 = 2.
It would be also possible to deal with different Lebesgue index on u r , u z and u θ .
Our paper is organized as follows. In the first section 2, we introduce the homogeneous hybrid Besov spaces (see also [2]) and recall the basis of Littlewood-Paley theory and the papraproduct. In section 3, we prove the theorem 1.2 and 1.3. In the last section, following the arguments of [13] we study the zero Mach limit of the solution when ε goes to 0 using Strichartz estimates in order to deal with the irrotationnal part which is governed roughly speaking by a wave equation.
2 Homogeneous and hybrid Besov spaces

Homogeneous Besov space and paraproduct
As usual, the Fourier transform of u with respect to the space variable will be denoted by F(u) or u. To build the Littlewood-Paley decomposition, we need to fix a smooth radial function χ supported in (for example) the ball B(0, 4 3 ), equal to 1 in a neighborhood of B(0, 3 4 ) and such that r → χ(r.e r ) is nonincreasing over R + . So that if we define ϕ(ξ) = χ(ξ/2)−χ(ξ), then ϕ is compactly supported in the annulus {ξ ∈ R d , 3 4 ≤ |ξ| ≤ 8 3 } and we have that, Then we can define the dyadic blocks (∆ l ) l∈Z by ∆ l := ϕ(2 −l D) (that is ∆ l u = ϕ(2 −l ξ) u(ξ)) so that, formally, we have u = We then define the space B s p,r as the subset of distributions u ∈ S h such that u Ḃs p,r is finite.
Once more, we refer to [2] (chapter 2) for properties of the inhomogeneous and homogeneous Besov spaces.
In this paper, we shall mainly work with functions or distributions depending on both the time variable t and the space variable x. We shall denote by C(I; X) the set of continuous functions on I with values in X. For p ∈ [1, ∞], the notation L p (I; X) stands for the set of measurable functions on I with values in X such that t → f (t) X belongs to L p (I).
The Littlewood-Paley decomposition enables us to work with spectrally localized (hence smooth) functions rather than with rough objects. We naturally obtain bounds for each dyadic block in spaces of type L ρ T L p . Going from those type of bounds to estimates in L ρ TḂ s p,r requires to perform a summation in r (Z). When doing so however, we do not bound the L ρ TḂ s p,r norm for the time integration has been performed before the r summation. This leads to the following notation: One can then define the space L σ The above operator T is called "paraproduct" whereas R is called "remainder". The decomposition (2.19) has been introduced by Bony in [3]. In this article we will frequently use the following estimates (we refer to [2] section 2.6, [10], for general statements, more properties of continuity for the paraproduct and remainder operators, sometimes adapted to L σ TḂ s p,r spaces): under the same assumptions there exists a constant C > 0 such that if 1/p 1 + 1/p 2 = 1/p, and 1/r 1 + 1/r 2 = 1/r: (2.20) Let us now turn to the composition estimates. We refer for example to [2] (Theorem 2.59, corollary 2.63)):

If u and v ∈Ḃ
Let us end this section by recalling the following estimates for the heat equation: ). Let u be a solution of: with A = µ∆ + (mu + λ)∇div. Then there exists C > 0 depending only on N, µ, ρ 1 and ρ 2 such that: .
If in addition r is finite then u belongs to C([0, T ],Ḃ s p,r ).

Hybrid Besov spaces
As explained, in the compressible Navier-Stokes system, the density fluctuation has two distinct behaviours in some low and high frequencies, separated by a frequency threshold. This leads to the definition of the hybrid Besov spaces. Let us begin with the spaces that are introduced by R. Danchin in [8,10] We will note in the sequel: .
Let us start now by recalling some estimates in Besov space for the following system: with (F , G ) external forces. More precisely in [21,10,7] it has been proved the following proposition by some different methods. In particular in [21], we introduce the notion of effective velocity which enables us to diagonalize the system (2.23).
the solution of (2.23). There exists a constant C depending only on µ, N , s and s such that the following estimate holds: . with: ds.

Dispersive estimate
Following B. Desjardins and E. Grenier in [15], we shall use some dispersive inequalities for the wave equation: the so-called Strichartz estimates (see [34,19,26] and the references therein). Here is the statement that we need. The reader may refer to [10] for a proof.
Proposition 2.4 Let (b, d) be a solution of the following system: Then for any s ∈ R and positive T (possibly infinite), the following estimate holds: ) and (r, p, N ) = (2, ∞, 3). Now, we can expect proposition 2.4 combined with estimates for the heat equation to provide us with uniform estimates for (b ε , u ε ), thus uniform bounds for the right-hand side of (W ε ). According to proposition 2.4, this should give us some convergence result. No further arguments are needed to get global convergence in the small in critical spaces,or local convergence in the large for more regular data. In the next two sections, we shall develop these ideas and give complete statements of our convergence results.

Proof of the theorem 1.2 and 1.3
Let us start with the proof of the theorem 1.2, the proof of theorem 1.3 will follow the same lines. We fix now the cute off frequency in (2.4) which is equal now to l 0 .

Proof of the existence
Our goal now consists in proving the existence of global strong solution for the system (1.5) with ε = 1 under the smallness condition (1.11). Let us briefly explain the main arguments of the proof.
1. We smooth out the data and get a sequence of local solutions (b n , u 1 n , u 2 n ) n∈N on [0, T n ] to (1.5) by using the result of [24].
2. We prove uniform estimates on (b n , u 1 n , u 2 n ) on [0, T n ] in E p 1,ν by using the proposition 2.3 and by studying the behavior of u 1 n in low and high frequencies. We deduce then that T n = +∞ .
3. We use compactness arguments to prove that the sequence (b n , u 1 n , u 2 n ) converges, up to extraction, to a solution of (1.5).

Construction of approximate solutions
We smooth out the data as follows: We observe that (u 1 0 ) n and (u 2 0 ) n are alway invariant by rotation since the Fourier transform conserves the property of radial functions and since we work with ϕ a radial function in section 2.

Uniform bounds
In this part, we aim at getting uniform estimates on (b n , u 1 n , u 2 n ) in the following space F T with the norm · F T : Let us start by estimating the incompressible part u 2 p,2 ). We observe that u 2 n verify the following equation: According to proposition 2.2, we have: .
(3.28) 14 with: (3.29) Using proposition 2.3 and 2.2, we have the following estimate on (b n , u 1 n , u 2 Therefore, it is only a matter of proving appropriate estimates for F 1 n and F 2 n by using proposition 2.20. Let us start with F 1 n , we have according to the proposition 2.20: .

(3.36)
For the remainder term we have: . (3.37) We proceed similarly in high frequencies, and using (3.35), (3.36) and (3.37) we get: . (3.38) Finally combining (3.32), (3.33) and (3.38) we obtain that: (3.39) By a basic bootstrap argument we deduce that the sequence (b n , u 1 n , u 2 n ) n∈N is uniformly bounded in n in F Tn . Since (∇u 1 n , ∇u 2 n ) is uniformly bounded in L 1 ,p,2 ) we deduce by using classical continuation criterion (see [11,24]) that for any n, T n = +∞. It is now classical to prove that the sequence (b n , u 1 n , u 2 n ) n∈N converges in the sense of the distributions to a solution (b, u 1 , u 2 ) of (1.5) (see for more details [2]). The uniqueness of the solution is also classical (see [2]).
The proof of the theorem 1.3 follows exactly the same lines. We are now going to prove the theorem 1.4 by following essentially the arguments of the proof of Danchin and He in [13].

Global uniform estimates in ε
Let us make the following change of functions c(t, x) = εb ε (ε 2 t, εx) and v(t, x) = εu ε (ε 2 t, εx). (4.40) Then (c, v) must verify: (4.41) Using the theorem 1.2, 1.3 and by scaling invariance, we know that there exists η > 0 such that if: then there exists a global strong solution (c, v) for the system (4.41). In addition we have:

The incompressible limit, weak convergence
Using the uniform estimates (4.42), we can show that the limit of (b ε , Qu ε ) when ε goes to 0 converges to 0 in the sense of the distribution. Similarly Pu ε converges to v in the sense of the distribution. The proof follows the same line than [13] (see also [33]).
Coming back to the original variables in (4.40) we get: We have by using (1.16) and the definition of the cut-off in (2.21): Similarly we have: The case N = 2: As previously we have: Taking ε = ν = 1 for the moment and using again proposition (2.4) to (1.3) we have: We take now r such that 1 r = c( 1 2 − 1 p 1 ). We have seen in the previous section that: Using interpolation as in the previous case, we deduce that for α < 0: .
Using the theorem 1.1 and the smallness assumption, we deduce that: which concludes the proof of the case N = 3.