On the regularity of solutions of the 3D Axisymmetric Navier-Stokes Equations with swirl L´eo Ag´elas

To cite this version: L´eo Ag´elas. On the regularity of solutions of the 3D Axisymmetric Navier-Stokes Equations with swirl. 18 pages. 2016.

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On the regularity of solutions of the 3D Axisymmetric Navier-Stokes Equations with swirl L´eo Ag´elas

∗

June 1, 2016

Abstract One of the most challenging questions in fluid dynamics is whether the incompressible threedimensional (3D) Navier-Stokes equations can develop a finite-time singularity from smooth and bounded initial data. It is well-known that global regularity of the incompressible Navier-Stokes equations is still wide open even in the axisymmetric case with general non-trivial swirl, although this case appeared more tractable than the full three-dimensional problem due to special features. In this paper, we prove that the blowup of the solutions of the 3D Navier-Stokes equations in the axisymmetric case with general non-trivial swirl can not occur at the time T if the scale-invariant quantity inf sup kΓ(t)1{r≤R} kL∞ is sufficiently small, where Γ = ruθ . To get R>0 t∈[0,T [

our result, we use some results of recent works on the stabilizing effect of the convection term in the 3D incompressible Navier-Stokes equations and the interaction between the swirling velocity and the angular vorticity fields. We show also that our regularity criterion is less restrictive than those involved in the recent papers. Keywords Navier-Stokes equations; 3D Axisymmetric flows; Regularity criterion Mathematics Subject Classification 35Q30, 76D03, 76D05

1

Introduction

The study of the incompressible Navier-Stokes in three space dimensions has a long history. For a long time ago, a global weak solution u ∈ L∞ (0, ∞; L2 (R3 ))3 and ∇v ∈ L2 (R3 ×(0, ∞))3 to the Navier-Stokes equations (2)-(3) was built by Leray [34]. In particular, Leray introduced a notion of weak solutions for the Navier-Stokes equation, and proved that, for every given v0 ∈ L2 (R3 )3 , there exists a global weak solution u ∈ L∞ ([0, +∞[; L2 (R3 ))3 ∩ L2 ([0, ∞[; H˙ 1 (R3 ))3 . Hopf has proved the existence of a global weak solution in the general case Rd , d ≥ 2, [23]. Several ways are known to construct weak solution ([15, 18, 11]), but the regularity and the uniqueness of this weak solution remained yet open in the general case, till now in spite of great efforts made (see [10, 12, 41, 36, 7, 46, 52, 12, 24, 47, 48, 51, 1, 17]). In two dimensions, the existence of classical solutions has been known for a long time ago (see [27, 37, 35, 49]). Thus a natural question, namely what can be said about the 3D axisymmetric flow, appears. Axisymmetric flow is an important subject in fluid dynamics and has become standard textbook material as a starting point of theoretical study for complicated flow patterns. Although the number of independent spatial variables is reduced by symmetry, some of the essential features and complexities of generic 3D flows remain. For example, when the swirling velocity is nonzero, ∗

Department of Mathematics at IFP Energies nouvelles, 1 & 4 avenue Bois Pr´eau, 92852 Rueil-Malmaison Cedex, France ([email protected])

1

there is a vorticity stretching term present. The first results in the existence of classical solutions were obtained in the late sixties for 3D axisymmetric flow without swirl (see [28], [50]) and later also in [33]. In the case of 3D axisymmetric flow with swirl, the question of finite time blow-up of solutions remained a challenging open problem in spite of tremendous efforts made (see [6, 42, 43, 8, 9, 21, 45, 22, 29, 16, 30, 26]). In several recent papers ([19, 20, 21, 22]), two systems of equations are proposed in order to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. By exploiting the special structure of the nonlinearity of the equations, the authors prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data. Furthermore, in more recent activities, regularity results for axi-symmetric solutions of the 3D Navier-Stokes are obtained under the assumption that some scale-invariant quantities remain finite (but not necessarly small). Indeed in [9, 8] it was proven that suitable axially symmetric solutions bounded by Cr −1+ǫ (t0 − ǫ t)− 2 with 0 ≤ ǫ ≤ 1 are smooth at time t0 , here r is the distance from a point x to the z-axis. Similar results were also obtained in [29, 30] and a local version in [45]. In [31] it was proven that there exists a constant C > 1 such that if there exists R ∈ [0, 12 ] such that sup kΓ(t)1{r≤R} kL∞ ≤ C1 | ln R|−2 then the solutions of the 3D Navier-Stokes equations t∈[0,T [

in the axisymmetric case with general non-trivial swirl and a viscosity ν of one can not blow up at the time T , where Γ(x, t) = ruθ (r, t), here uθ is the swirl component of u and r = |x′ | with x′ ∈ R2 such that x ≡ (x′ , z) ∈ R3 . Later in [53], the previous result have been improved in the sense that if there exists R ∈ 3 [0, 21 ] such that sup kΓ(t)1{r≤R} kL∞ ≤ C1 | ln R|− 2 then the solutions of the 3D Navier-Stokes t∈[0,T [

equations in the axisymmetric case with general non-trivial swirl and a viscosity ν of one can not blow up at the time T . In this paper, from our Theorem 5.1, we obtain that the blowup of the solutions of the 3D Navier-Stokes equations in the axisymmetric case with general non-trivial swirl and a viscosity ν of one can not occur at the time T if the scale-invariant quantity inf sup kΓ(t)1{r≤R} kL∞ is R>0 t∈[0,T [

smaller than a certain absolute constant. We draw attention to the fact that our regularity criterion is less restrictive than those involved in [8, 9, 29, 30, 45], indeed under their assumptions we infer that Γ(x, t) is H¨older continuous at (r, t) ≡ (0, T ) uniformly (see section 5 in [9], Theorem 3.1 in [8], see also Theorem 1.1 for [30]). Then, for any ǫ > 0, we infer that there exists tǫ ∈ [0, T [ and Rǫ > 0 such that for all t ∈ [tǫ , T [ ǫ ǫ ˜ǫ = and 0 < R ≤ Rε , k(Γ(t) − Γ(tǫ ))1{r≤R} kL∞ ≤ and by setting R we 2 2(1 + kukL∞ (R3 ×[0,tǫ ]) ) ˜ ǫ , kΓ(t)1{r≤R} kL∞ ≤ R ˜ ǫ ku(t)kL∞ ≤ ǫ . Then by get that for all t ∈ [0, tǫ ] and for all 0 < R ≤ R 2 ¯ ǫ = min{Rǫ , R ˜ ǫ }, we infer that for all t ∈ [0, T [, kΓ(t)1 taking R ¯ǫ } kL∞ ≤ ǫ and then we infer {r≤R that for any ǫ > 0, inf sup kΓ(t)1{r≤R} kL∞ ≤ ǫ which means that inf sup kΓ(t)1{r≤R} kL∞ = R>0 t∈[0,T [

R>0 t∈[0,T [

0. Then, we conclude that the regularity criteria involved in [8, 9, 29, 30, 45] imply that inf sup kΓ(t)1{r≤R} kL∞ = 0 which prove that their regularity criteria are more restrictive

R>0 t∈[0,T [

than our criterion. We draw also attention to the fact that our regularity criterion is less restrictive than those involved in [31, 53] since to get non blowup of the solutions, we require only that there exists R > 0 such that sup kΓ(t)1{r≤R} kL∞ ≤ γ0 where γ0 > 0 is an absolute constant. t∈[0,T [

Moreover, our criterion is bounded by kΓ0 kL∞ thanks to (17), this feature eases the numerical detection of potential blowup of the solutions.

2

To obtain this result, we have been able to show the following energy estimate on [0, T [:     3 1 8 d 1 3 2 2 ku1 (t)k3 + kω1 (t)k2 − C(1 + kΓ(t)χ{r≤R} kL∞ )kΓ(t)χ{r≤R} kL∞ k∇|u1 (t)| 2 k22 + dt 3 2 9
kΓ0 kL∞ 1 1 + kΓ0 k2L∞ ku1 (t)k33 , + k∇ω1 (t)k22 ≤ C 2 2 R (1) ωθ uθ and Γ = ruθ . Then, the paper is organized as follows: where u1 = , ω1 = r r • In section 2, we recall some results known about the solutions of Navier-Stokes equations. • In section 3, we introduce the 3D axisymmetric incompressible Navier-Stokes equations with some known results. • In section 4, we recall some estimates on Γ. • In section 5, we obtain an estimate on ku1 (t)k33 + kω1 (t)k22 in Lemma 5.4 by showing inequality (1) and then we obtain our Theorem 5.1. First, we give some notations. Some notations : For any m ∈ N∗ function ϕ defined on Rm × [0, +∞[, for all t ≥ 0, we denote by ϕ(t) the function defined on Rm by xv 7−→ ϕ(x, t). For any vector x = (x1 , x2 , x3 ) ∈ R3 , we u 3 uX denote by |x| the norm defined by |x| = t x2 . For any axisymmetric function f defined on i

i=1

3 R3 , for the sake of simplicity, the p value f (x) with x = (x, y, z) ∈ R is denoted using coordinates d 2 2 cylindrical, f (r, z) with r = x + y . For any d ≥ 1, Ω ⊂ R , we denote by C0∞ (Ω) (resp C0 (Ω)) the space constituted by all infinitely differentiable (resp continuous) functions with compact support in Ω. For any Ω ⊂ Rd , with d ≥ 1, we denote by χΩ , the function defined on Rd , by χΩ (x) = 1 for all x ∈ Ω and 0 elsewhere. For any R > 0, we denote by χ{r≤R} (resp χ{r≥R} ) the function defined on R+ × R such that for all (r, z) ∈ZR+ × R, χ{r≤R} (r, z) = 1 (resp

χ{r≥R} (r, z) = 1) for all r ≤ R and 0 elsewhere. The symbol denotes the integral over R3 Z ∞ Z ∞ Z 2π ... r dθ dz dr. For any q > 1, the norm in equal using cylindrical coordinates to 0

−∞

0

Lq (R3 ) will be denoted by k · kLq and also k · kq . We denote A . B, the estimate A ≤ C B where C > 0 is an absolute constant.

2

Local regularity of solution of Navier-Stokes equation

In this section, we deal with the main result on local regularity of Navier-Stokes equations in its general form. Consider the Navier-Stokes equations, ( ∂u + (u · ∇)u − ν∆u + ∇p = 0, (2) ∂t ∇ · u = 0, in which u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)) ∈ R3 , p = p(x, t) ∈ R and ν > 0 denote respectively the unknown velocity field, the scalar pressure function of the fluid at the point (x, t) ∈ R3 × [0, ∞[ and ν > 0 the viscosity of the fluid, with initial conditions, u(x, 0) = u0 (x) for a.e x ∈ R3 . 3

(3)

Without loss of generality, in what follows, we assume that ν = 1. Assuming u0 ∈ H m (R3 ) for a given m ≥ 1, thanks to the results obtained in [34], Theorem 3.5 in [25], Lemma 5.6 [11], Theorem 6.1 [12] or the results obtained in [18], we get that there exists a time T > 0 such that there exists an unique solution u ∈ C([0, T [; H m (R3 ))3 ∩ L2 ([0, T [; H m+1 (R3 ))3 to the Navier-Stokes Equations (2)-(3). Due to the regularity of solution of Navier-Stokes equation, u ∈ C([0, T [; H m (R3 ))3 and thanks to the results obtained in [44], [36], we get the energy equality, in other words, for all t ∈ [0, T [, Z t 2 (4) k∇uk2L2 (R3 )3×3 = ku0 k2L2 (R3 ) . ku(t)kL2 (R3 )3 + 2 0

Moreover if u 6∈ C([0, T ]; H m (R3 ))3 , then thanks to the results obtained in [34], Theorem 6.1 [11], Lemma 6.2 [12], we infer that, lim sup k∇u(t)kL2 (R3 )3×3 = +∞,

(5)

t→T

and thanks to Theorem 3.1.1 in [3], we have also, lim sup kω(t)kL2 (R3 )3 = +∞,

(6)

t→T

where ω = ∇ × u is the vorticity of u. Moreover up to the initial time, the solution of Navier-Stokes equation is smooth, u ∈ C ∞ (R3 ×]0, T [) (see Theorem 3 and 4 in [18], see also Lemma 5.6 and Theorem 5.2 in [11]). We denote by ω0 = ∇ × u0 the vorticity of u0 .

3

Axisymmetric flows

By an axisymmetric solution of the Navier-Stokes equations, we mean a solution of the equations of the form u(x, y, z, t) = ur (r, z, t)er + uθ (r, z, t)eθ + uz (r, z, t)ez . in the cylindrical coordinate system, where we used the basis p x y y x er = ( , , 0), eθ = (− , , 0), ez = (0, 0, 1) and r = x2 + y2 r r r r In the above expression, uθ is called the swirl component of the velocity field u. For the axisymmetric solutions, we can rewrite the equations (2) as follows :  ∂uθ ∂uθ ur ∂uθ  + ur + uz = L uθ − uθ ,    ∂t ∂r ∂z r   ∂ur u2θ  ∂ur ∂ur + ur + uz = L ur + + ∂r p, (7) ∂t ∂r ∂z r  ∂uz ∂uz ∂uz    + ur + uz = ∆uz + ∂z p,   ∂r ∂z  ∂t ∂r (rur ) + ∂z (ruz ) = 0.

For the axisymmetric vector field u, we can compute the vorticity ω = ∇ × u as follows, ω = ωr er + ωθ eθ + ωz ez , where ωr = −(uθ )z , ωθ = (ur )z − (uz )r and ωz = 1r (ruθ )r . Moreover, the vorticity components satisfy :  ∂ωθ ∂ωθ ur 2 ∂ωθ   + ur + uz = L ω θ + ω θ − uθ ω r ,   ∂t ∂r ∂z r r  ∂ωr ∂ωr ∂ur ∂ur ∂ωr + ur + uz = L ωr + ωr + ωz ,  ∂t ∂r ∂z ∂r ∂z     ∂ωz + ur ∂ωz + uz ∂ωz = ∆ωz + ∂uz ωz + ∂uz ωr . ∂t ∂r ∂z ∂z ∂r 4

(8)

The operator L and ∆ is defined by : 1 = ∂r2 + ∂r + ∂z2 , r   1 = ∆− 2 . r

∆ L

(9)

One can derive evolution equations for (uθ , ωθ , ψθ ) which completely determine the evolution of the three-dimensional axisymmetric Navier-Stokes equations (7) once the initial condition is given (see e.g. [40], [6]) :  ∂uθ ∂uθ ur ∂uθ   + u + u = L u − uθ , r z θ   ∂t ∂r ∂z r 2 ∂ωθ ∂ωθ ∂ωθ 1 ∂uθ ur (10) + ur + uz = L ωθ + + ωθ ,    ∂t ∂r ∂z r ∂z r  −L ψθ = ωθ where ur and uz can be expressed in terms of the angular component of the stream function ψθ as follows : ∂ψθ 1 ∂(rψθ ) ur = − , uz = . (11) ∂z r ∂r We note that the incompressibility condition implies that, ∂r (rur ) + ∂z (ruz ) = 0.

(12)

In [39], there are shown the equivalence between the systems of equations (2) and (10)-(12), to mention their main result, we introduce some spaces with the same notations as in [39]. Denote by Csk the axisymmetric divergence free subspace of C k vector fields : Csk (R3 , R3 ) = {u ∈ C k (R3 , R3 )| ∂θ uz = ∂θ ur = ∂θ uθ = 0, ∇ · u = 0}. ′

Thanks to Lemma 2 (see also Lemma 2 ) in [39], we have, Csk (R3 , R3 ) = {ueθ + ∇ × (ψeθ )| u ∈ Csk (R+ × R), ψ ∈ Csk+1 (R+ × R)}, where Csk (R+ × R) is the function space defined by, Csk (R+ × R) = {f (r, z) ∈ C k (R+ × R)| ∂r2j f (0+ , z) = 0, 0 ≤ 2j ≤ k}. We can now define the Sobolev spaces for axisymmetric solenoidal vector fields : H˙ s1 (R+ × R) = Completion of Cs1 (R+ × R) ∩ C0 (R+ × R) with respect to k · kH˙ 1 (R+ ×R)

Hsk (R+ × R) = Completion of Csk (R+ × R) ∩ C0 (R+ × R) with respect to k · kH k (R+ ×R) , where C0 denotes the space of compactly supported functions. As mentionned in [22] and proved in [39], any smooth solution of the 3D axisymmetric NavierStokes equations must satisfy the following compatibility condition at r = 0 : uθ (0, z, t) = ψθ (0, z, t) = ωθ (0, z, t) = 0.

(13)

More precisely, we have the following result, thanks to Lemma 8, Theorem 4 and Corollary 3 in [39], Theorem 3.1. If u0 ∈ H k (R3 )3 is an axisymmetric solenoidal vector field with k ≥ 1, then there exists u0,θ ∈ Hsk (R+ × R), ψ0,θ ∈ H˙ s1 (R+ × R) with L ψ0,θ ∈ Hsk−1 (R+ × R) such that u0 = u0,θ eθ + ∇ × (ψ0,θ eθ ) and there exists a time T > 0 such that u = uθ eθ + ∇ × (ψθ eθ ) corresponds to the unique strong solution to the Navier-Stokes equations (2) in the class C([0, T [; H k (R3 )3 ) where (uθ , ψθ , ωθ ) is solution to (10)-(12) for the initial data (u0,θ , ψ0,θ , −L ψ0,θ ) and satisfies, ψθ ∈ C([0, T [; Hsk+1 (R+ × R)), uθ ∈ C([0, T [; Hsk (R+ × R)),

ωθ ∈ C(0, T [; Hsk−1 (R+ × R)). 5

4

Estimates for axisymmetric solution

In this section, we recall some estimates on the quantity Γ = ruθ . For this, it is assumed that u0 ∈ H m is a axisymmetric solenoidal vector field, with m ≥ 2, then Theorem 3.1 holds and there exists a time T > 0 such that there exists an unique strong solution u to the Navier-Stokes equations (2) which belongs to C([0, T [; H m (R3 ))∩L2 ([0, T [; H m+1 (R3 )) with m ≥ 2 (see Section 2). A special feature of the axisymmetric Navier-Stokes equations is that the quantity Γ = ruθ satisfies an parabolic equation on ]0, T [ with singular drift terms:   2 (14) ∂t + b · ∇ − ∆ + ∂r Γ = 0 r with boundary conditions, Γ|r=0 = 0,

(15)

Γ(x, 0) = Γ0 (x) for a.e x ∈ R3 ,

(16)

with initial conditions, where, Γ0 = ru0,θ , b = ur er + uz ez , b · ∇ = ur ∂r + uz ∂z and div b = 0.

Note that in equation (14), the convection term has absorbed the term

ur uθ in the first equation r

(10), which highlights the stabilizing effect of the convection. We remark also that Γ enjoys the maximal principle. Indeed thanks to inequality (4.6) in [42] (see also Proposition 1 in [6]), we have for all q ∈ [2, ∞], for all t ∈ [0, T [, kΓ(t)kLq (R3 ) ≤ kΓ0 kLq (R3 ) .

5

(17)

Global regularity

In this section, we assume that u0 ∈ H m is an axisymmetric solenoidal vector field, with m ≥ 2 and Γ0 = ru0,θ ∈ L2 (R3 ) ∩ L∞ (R3 ), then Theorem 3.1 holds and there exists a time T > 0 such that there exists an unique strong solution u to the Navier-Stokes equations (2) which belongs to C([0, T [, H m (R3 )) ∩ L2 ([0, T [; H m+1 (R3 )) (see Section 2). This section is devoted to the proof of Theorem 5.1. The proof of our Theorem is obtained in three steps : ur • First, thanks to the convection term, we eliminate an annoying term in (10), ωθ , by r using the change of unknowns from (uθ , ψθ , ωθ ) to (u1 , ψ1 , ω1 ) (see (18)). • Second, thanks to Lemmata 5.2 and 5.3 , we establish in Lemma 5.4 a dynamic control of ku1 (t)k33 + kω1 (t)k22 which reveals a dynamic interaction between the angular velocity and the angular vorticity fields. • Third, using this dynamic control, we obtain the proof of our Theorem 5.1. We re-write uθ and ψθ as follows : uθ (r, z, t) = ru1 (r, z, t),

ωθ (r, z, t) = rω1 (r, z, t),

ψθ (r, z, t) = rψ1 (r, z, t).

(18)

Since m ≥ 2, then u ∈ C([0, T [; H 2 (R3 ))3 ∩ L2 ([0, T [; H 3 (R3 ))3 and thanks to Lemmata 3-6 in [42], we deduce that, u1 ∈ C([0, T [; H 1 (R3 )) ω1 ∈ C([0, T [; L2 (R3 )) ∩ L2 ([0, T [; H 1 (R3 )).

(19)

Thanks to (19) and Lemma 1 in [22] used firstly with u = ψ1 , f = ω1 , secondly with u = ψ1 , f = ω1 and using the same choice of the weight w as in Lemma 2 ([22]), we get, ψ1 ∈ C([0, T [; H 2 (R3 )) ∩ L2 ([0, T [; H 3 (R3 )). 6

(20)

As in [21], from (10), we derive the following equivalent system for (u1 , ω1 , ψ1 ) :    3  2 2  ∂t u1 + ur ∂r u1 + uz ∂z u1 = 2u1 ∂z ψ1 (t) + ∂r u1 + ∂z u1 + ∂r u1 ,    r     3 2 2 2 ∂t ω1 + ur ∂r ω1 + uz ∂z ω1 = ∂z (u1 ) + ∂r ω1 + ∂z ω1 + ∂r ω1 ,  r      3    − ∂r2 ψ1 + ∂z2 ψ1 + ∂r ψ1 = ω1 r

(21)

where,

∂ψ1 1 ∂(r 2 ψ1 ) , uz = . (22) ∂z r ∂r Note that in the new system (21), the convection term has absorbed one of the vortex-stretching ur ωθ , which originally appears in the second equation of (10). In some sense, the conterms r vection term has already stabilized one of the potentially destabilized vortex-stretching terms in the above reformulation. To obtain the proof of the crucial Lemma 5.4, we use Lemma 5.2 and Lemma 5.3. Lemma 5.2 depends on Lemma 5.1 which is an immediate consequence of CKN-type inequalities proved in [5]. ur = −r

Lemma 5.1. There exists a constant C > 0 such that for all v ∈ C0∞ ([0, +∞[\{0}) and α > 12 , we have, Z ∞ Z ∞ 2 2(α−1) |v ′ (r)|2 r 2α dr. |v(r)| r dr ≤ C 0

0

Here is the proof of Lemma 5.2. Lemma 5.2. There exists a constant C > 0 such that for all t ∈ [0, T [ and for all R > 0, we have, 3

ku1 (t)k44 ≤ C kΓ(t)χ{r≤R} kL∞ k∇|u1 (t)| 2 k22 +

CkΓ0 kL∞ ku1 (t)k33 . R2

Proof. Let R > 0. Consider the cut-off function ζ defined on R+ for which 0 ≤ ζ ≤ 1, ζ = 1 on [0, 21 ], supp  r ζ ⊂ [0, 1]. Now, we consider the rescaled cut-off function ζR defined on R+ by ζR (r) = ζ . For any x ∈ R3 , we write x under the form x = (x′ , z) where x′ ∈ R2 . Then, R we have, Z Z 4 (|u1 (x, t)|ζR (|x′ |) + |u1 (x, t)|(1 − ζR (|x′ |)))4 dx |u1 (t)| = 3 R Z Z ( |u1 (x, t)|(1 − ζR (|x′ |)) )4 dx ( |u1 (x, t)|ζR (|x′ |) )4 dx + 4 ≤ 4 3 3 ZR ZR ′ 4 |u1 (x, t)|4 χ{|x′ |≥ R } dx. ( |u1 (x, t)|ζR (|x |) ) dx + 4 ≤ 4 R3

R3

2

With r = |x′ |, we recall that Γ = ruθ and Γ0 = ruθ (0), we notice that r 2 u1 (t) = Γ(t), then |Γ(t)|χ{r≥ R } 4|Γ(t)| 2 |u1 (t)|4 χ{r≥ R } = |u1 (t)|3 ≤ |u1 (t)|3 and thanks to (17), we obtain, 2 2 r R2 |u1 (t)|4 χ{r≥ R } ≤ 2

4kΓ0 kL∞ |u1 (t)|3 . R2

Therefore, we deduce for all t ∈ [0, T [, Z Z Z 16kΓ0 kL∞ ′ 4 4 ( |u1 (x, t)|ζR (|x |) ) dx + |u1 (t)| ≤ 4 |u1 (t)|3 . 2 R 3 3 R R 7

(23)

We consider u1 (x, t) under the form u1 (r, z, t), then we have, Z Z ∞Z ∞ (|u1 (x, t)|ζR (|x′ |))4 = 2π |u1 (r, z, t)|4 ζR (r)4 r dr dz. −∞

0

For a.e z ∈ R, thanks to Lemma 5.1 used with v = (|u1 (·, z, t)|ζR (r))2 and α = 23 , we obtain, Z ∞ Z ∞ 4 |∂r (|u1 (r, z, t)|ζR (r))2 |2 r 3 dr (|u1 (r, z, t)|ζR (r)) r dr ≤ C 0 0 Z ∞ (|u1 (r, z, t)|ζR (r))2 |∂r (|u1 (r, z, t)|ζR (r))|2 r 3 dr = 4C Z0 ∞ |Γ(r, z, t)| |u1 (r, z, t)| |∂r (|u1 (r, z, t)|ζR (r))|2 r dr ≤ 4C Z0 ∞ ′ ≤ 8C |Γ(r, z, t)| |u1 (r, z, t)|(|u1 (r, z, t)|2 ζR (r)2 + ζR (r)2 |∂r |u1 (r, z, t)||2 ) rdr 0   Z ∞ 3 kζ ′ k2L∞ 4 2 2| + χ |∂ |u (r, z, t)| r dr. |Γ(r, z, t)| |u1 (r, z, t)|3 ≤ 8C r 1 R2 9 {r≤R} 0 (24) Thanks to Inequality (17), then from (24), we infer that there exists a constant C1 > 0 such that, Z ∞ Z Z ∞ 3 kΓ0 kL∞ ∞ 3 4 |∂r |u1 (r, z, t)| 2 |2 rdr. |u1 (r, z, t)| rdr + C1 kΓ(t)χ{r≤R} kL∞ (|u1 (r, z, t)|ζR (r)) r dr ≤ C1 2 R 0 0 0 Therefore, we obtain, Z Z Z 3 kΓ0 kL∞ ′ 4 3 (|u1 (x, t)|ζR (|x |)) dx ≤ C1 |u1 (t)| + C1 kΓ(t)χ{r≤R} kL∞ |∇|u1 (t)| 2 |2 . (25) 2 R R3 Then, using (23) and (25), we conclude the proof. To prove Lemma 5.4, the main Lemma in this section, we need Lemma 5.3. Lemma 5.3. There exists a constant C > 0 such that for all f ∈ L2 (R2 ) radial function such that |x|2 f ∈ L2 (R2 ) and g ∈ H 2 (R2 ), we have, Z

R

Z f g ≤ C 2

4

R2

2

|x| f (x) dx

1

2

k∆gkL2 (R2 ) .

Proof. Since f is a radial function, there exists ζ a real function on R+ such that for a.e x ∈ R2 , f (x) = ζ(|x|),

(26)

and using the change of variables with polar coordinates x = (r cos θ, r sin θ), r ∈ R+ and θ ∈ [0, 2π], we obtain, √ 1 kf kL2 (R2 ) = 2πkζ(r)r 2 kL2 (R+ ) , √ (27) 5 k |x|2 f kL2 (R2 ) = 2πkζ(r)r 2 kL2 (R+ ) . Let K > 0, ζK the real function defined on R+ by ζK (r) = ζ(r)χ{0≤r≤K} for all r ≥ 0. We introduce also φK the real function defined on R∗+ for all r > 0 by, Z ∞ Z ∞ 1 φK (r) = τ ζK (τ ) dτ dρ. (28) ρ r ρ Using successively the fact that supp ζK ⊂ [0, K] and |ζK | ≤ |ζ|, for all α ≥ 0 and for a.e τ > 0, we get, ! 1  1 1 1 1 K α+ 2  1 −α α+ −α α+ 2 |ζ(τ )| . (29) τ |τ ζK (τ )| = τ 2 |τ 2 ζK (τ )| ≤ τ 2 K 2 |ζ(τ )| = τα 8

Using definition (28), inequality (29), Cauchy-Schwarz inequality and (27), we deduce that φK ∈ C 1 (]0, +∞[) and for all r > 0 and α > 12 , 1

K α+ 2 p kf kL2 (R2 ) , (α − 12 ) 2π(2α − 1) 1 K α+ 2 α+ 21 ′ |φK (r)| ≤ p kf kL2 (R2 ) , r 2π(2α − 1) 1 1 r α |(rφ′K (r))′ | ≤ K α+ 2 |r 2 ζ(r)|. 1

r α− 2 |φK (r)|

Let us show that

≤

1

(30)

3

r 2 φK ∈ L2 ([0, +∞[) and r 2 φ′K ∈ L2 ([0, +∞[).

(31)

1 3 and α = 2, we infer respectively that r 4 |φK (r)| ≤ 4 3 and r 2 |φK (r)| ≤ CK kf kL2 (R2 ) , where CK > 0 is a real depending only on K. Then,

Using the first inequality of (30) with α = CK kf kL2 (R2 ) we get

Z

+∞

Z

1

Z

+∞

rφK (r)2 dr Z +∞ Z 1 1 1 1 3 (r 2 φK (r))2 dr r 2 (r 4 φK (r))2 dr + = 2 r 1 0 5 2 2 ≤ C kf kL2 (R2 ) . 3 K

2

rφK (r) dr = 0

2

rφK (r) dr +

1

0

1

3

Therefore, we deduce that r 2 φK ∈ L2 ([0, +∞[). It remains to show that r 2 φ′K ∈ L2 ([0, +∞[). 5 3 Using the second inequality of (30) with α = and α = 2, we infer respectively that r 4 |φ′K (r)| ≤ 4 eK kf kL2 (R2 ) and r 52 |φ′ (r)| ≤ C eK kf kL2 (R2 ) , where C eK > 0 is a real depending only on K. Then, C K we get Z +∞ Z 1 Z +∞ 3 ′ 3 ′ 2 2 r |φK (r)| dr = r |φK (r)| dr + r 3 |φ′K (r)|2 dr 0 0 1 Z +∞ Z 1 5 1 1 5 ′ ′ 2 2 4 (r 2 φK (r))2 dr r (r φK (r)) dr + = 2 r 1 0 5 e2 2 C kf kL2 (R2 ) . ≤ 3 K 3

Therefore, we deduce that r 2 φ′K ∈ L2 ([0, +∞[).

By using also the third inequality of (30) with α =

3 and thanks to (27), we infer, 2

3

r 2 (rφ′K )′ ∈ L2 ([0, +∞[).

(32)

3 Then thanks to (31) and (32), by using twice Lemma 5.1 with α = , we deduce, 2 3

1

kr 2 φK kL2 (R+ ) . kr 2 φ′K kL2 (R+ ) 1 = kr 2 (rφ′K )kL2 (R+ ) 3 . kr 2 (rφ′K )′ kL2 (R+ ) 5 ˜ K (r)kL2 (R ) , = kr 2 ∆φ +

(33)

˜ K (r) := 1 (rφ′K )′ . From (28), we notice, where for all r > 0, ∆φ r ˜ K (r) = ζK (r). ∆φ 9

(34)

Then, from (33), using (34), we obtain, 5

1

kr 2 φK kL2 (R+ ) . kr 2 ζK (r)kL2 (R+ ) 5 ≤ kr 2 ζ(r)kL2 (R+ ) .

(35)

We introduce the radial function ΦK defined on R2 by, ΦK (x) = φK (|x|).

(36)

˜ K (|x|) and thanks to (34), we have Then, we get ∆ΦK (x) = ∆φ ∆ΦK (x) = ζK (|x|) = ζ(|x|)χ{|x|≤K} = f (x)χ{|x|≤K} . Since, we have, Z

R

f g 2

Then, we deduce, Z

R

Z Z f (x)χ{|x|>K} g(x) dx = f (x)χ{|x|≤K} g(x) dx + 2 2 Z R ZR f (x)χ{|x|>K}g(x) dx . ≤ f (x)χ{|x|≤K} g(x) dx + R2

R2

Z f g ≤ 2

R

Z ∆ΦK (x)g(x) dx + 2

R

f (x)χ{|x|>K} g(x) dx . 2

(37)

For the first term at the right hand side of inequality (37), using integration by parts and thanks to Cauchy-Schwarz inequality, we get, Z Z 2 ∆ΦK (x)g(x) dx = 2 ΦK (x)∆g(x) dx (38) R R ≤ kΦK kL2 (R2 ) k∆gkL2 (R2 ) . Using the change of variables with polar coordinates, from (36), we observe, √ 1 kΦK kL2 (R2 ) = 2πkφK (r)r 2 kL2 (R+ ) , then thanks to (35) and (27), we deduce, kΦK kL2 (R2 ) . k |x|2 f kL2 (R2 ) .

Then, using (39), from (38), we deduce, Z ∆ΦK (x)g(x) dx . k |x|2 f kL2 (R2 ) k∆gkL2 (R2 ) .

(39)

(40)

R2

For the second term at the right hand side of inequality (37), thanks to Cauchy-Schwarz inequality, we obtain, Z ≤ kf kL2 ({x∈R2 ,|x|>K})kgkL2 (R2 ) . f (x)χ g(x) dx (41) {|x|>K} 2 R

Using (40) and (41), from (37), we obtain, Z f g . k |x|2 f kL2 (R2 ) k∆gkL2 (R2 ) + kf kL2 ({x∈R2 ,|x|>K})kgkL2 (R2 ) .

(42)

R2

Since f ∈ L2 (R2 ), then kf kL2 ({x∈R2 ,|x|>K}) → 0 as K → ∞. Then, taking the limit in inequality (42) as K → ∞, we obtain, Z f g . k |x|2 f kL2 (R2 ) k∆gkL2 (R2 ) , R2

which concludes the proof.

10

Now, we turn to the proof of the main Lemma of this section. Lemma 5.4. There exist two absolute constants γ0 > 0 and C > 0 such that if there exists R > 0 such that, sup kΓ(t)χ{r≤R} kL∞ ≤ γ0 ,

t∈[0,T [

then we get that for all t ∈ [0, T [,    
1 1 1 kΓ0 kL∞ 1 3 2 3 2 2 ku1 (t)k3 + kω1 (t)k2 ≤ ku1 (0)k3 + kω1 (0)k2 exp 3C 1 + kΓ0 kL∞ t . 3 2 3 2 R2

Proof. We multiply the first equation of (21) by u1 (t) |u1 (t)|, integrate it over R3 , use the incompressibility condition (12) and integration by parts, to obtain for all t ∈ [0, T [, Z Z Z 3 2 ∞ 1 d 8 2 3 3 ku1 (t)k3 + |u1 (0, z, t)| dz = 2 |u1 (t)|3 ∂z ψ1 (t). (43) |∇|u1 (t)| 2 | + 3 dt 9 3 −∞ Note that, in order to treat the convective term, we have integrated by parts and the boundary integrals have vanished at r = 0 due to the fact that uθ (0, z, t) = 0, while near r = ∞ due to the standard density argument. We observe,  Z Z Z ′ 3 ′ ′ 3 |u1 (x , z, t)| ∂z ψ1 (x , z, t)dx dz. |u1 (t)| ∂z ψ1 (t) = (44) R

R2

Thanks to (19), (20) and Lemma 5.3, there exists a constant C0 > 0 such that for a.e z ∈ R, Z |u1 (x′ , z, t)|3 ∂z ψ1 (x′ , z, t)dx′ ≤ C0 k |x′ |2 |u1 |3 (·, z, t)kL2 (R2 ) k∇2x′ (∂z ψ1 )(·, z, t)kL2 (R2 ) . (45) R2

From (44), thanks to (45) and Cauchy-Schwarz inequality, we get, Z

3

|u1 (t)| ∂z ψ1 (t) ≤ C0

Z

′ 2

R

3

k |x | |u1 |

(·, z, t)k2L2 (R2 )

dz

 1 Z 2

R

= C0 k |x′ |2 |u1 (t)|3 kL2 (R3 ) k∇2x′ ∂z ψ1 (t)kL2 (R3 ) .

k∇2x′ (∂z ψ1 )(·, z, t)k2L2 (R2 )

1 2

dz

(46) Thanks to Lemma 1 in [22] used with u = ∂z ψ1 (t), f = ∂z ω1 (t) and using the same choice of the weight w as in Lemma 2 of [22], we deduce that there exists a constant C1 > 0 such that for all t ∈ [0, T [, Z Z (47) |∇2 ∂z ψ1 (t)|2 ≤ C1 |∂z ω1 (t)|2 .

Then, thanks to (46) and (47), we deduce that there exists a real C2 > 0 such that for all t ∈ [0, T [, Z 1 Z 2 3 ′ 2 3 2 2 |u1 (t)| ∂z ψ1 (t) ≤ C2 k |x | |u1 (t)| kL2 (R3 ) |∂z ω1 (t)| . (48)

Recalling Γ = ruθ , with r = |x′ |, we notice that |x′ |2 u1 (t) = Γ(t), then |x′ |2 |u1 (t)|3 = |Γ(t)| |u1 (t)|2 , then from (48), we obtain, Z 2 |u1 (t)|3 ∂z ψ1 (t) ≤ C2 k |Γ(t)| |u1 (t)|2 k2 k∂z ω1 (t)k2 (49) 1 ≤ C22 k |Γ(t)| |u1 (t)|2 k22 + k∂z ω1 (t)k22 . 4

11

Further, we have k |Γ(t)|

|u1 (t)|2 k22

=

Z

ZR

3

|Γ(x, t)|2 |u1 (x, t)|4 dx

Z

|Γ(x, t)χ{|x′ |>R} |2 |u1 (x, t)|4 dx Z Z kΓ0 k3L∞ |Γ(x, t)χ{|x′ |≤R} |2 |u1 (x, t)|4 dx + |u1 (x, t)|3 dx, ≤ 2 R 3 3 R R

=

R3

2

4

|Γ(x, t)χ{|x′ |≤R} | |u1 (x, t)| dx +

R3

where for the last inequality we have used the fact that |Γ(x, t)χ{|x′ |>R} |2 |u1 (x, t)|4 = |Γ(x, t)χ{|x′ |>R} |2 ≤ ≤

|Γ(x, t)| |u1 (x, t)|3 |x′ |2

kΓ(t)k3L∞ |u1 (x, t)|3 R2 kΓ0 k3L∞ |u1 (x, t)|3 ( thanks to (17)). R2

Then, from (49), we obtain Z kΓ0 k3L∞ 1 ku1 (t)k33 + k∂z ω1 (t)k22 . 2 |u1 (t)|3 ∂z ψ1 (t) ≤ C22 kΓ(t)χ{r≤R} k2L∞ ku1 (t)k44 + C22 2 R 4 (50) Using (50), from (43), we deduce that for all t ∈ [0, T [,

3 kΓ0 k3L∞ 1 d 1 8 ku1 (t)k33 + k∇|u1 (t)| 2 k22 ≤ k∂z ω1 (t)k22 +C22 kΓ(t)χ{r≤R} k2L∞ ku1 (t)k44 +C22 ku1 (t)k33 . 3 dt 9 4 R2 (51) 3 We multiply the first equation of (21) by ω1 (t), integrate it over R , use the incompressibility condition (12), then we obtain for all t ∈ [0, T [, Z Z Z ∞ 1d |ω1 (0, z, t)|2 dz = ω1 (t)∂z (u1 (t)2 ). (52) kω1 (t)k22 + |∇ω1 (t)|2 + 2 dt −∞

By using integration by parts, Cauchy-Schwarz inequality and Young inequality, we deduce that for all t ∈ [0, T [, Z Z 2 ω1 (t)∂z (u1 (t) ) = − ∂z ω1 (t)u1 (t)2

≤ k∂z ω1 (t)k2 ku1 (t)k24 1 ≤ k∂z ω1 (t)k22 + ku1 (t)k44 . 4 Using (53), from (52), we obtain for all t ∈ [0, T [, 1 d 3 kω1 (t)k22 + k∇ω1 (t)k22 ≤ ku1 (t)k44 . 2 dt 4

(53)

(54)

We sum inequalities (51) and (54), then, we obtain for all t ∈ [0, T [,   3 1 8 1 d 1 3 2 ku1 (t)k3 + kω1 (t)k2 + k∇|u1 (t)| 2 k22 + k∇ω1 (t)|22 ≤ (1 + C22 kΓ(t)χ{r≤R} k2L∞ )ku1 (t)k44 dt 3 2 9 2 kΓ0 k3L∞ +C22 ku1 (t)k33 . R2 (55) Thanks to Lemma 5.2 and inequality (17), from (55), we deduce that there exists a constant C3 > 0 such that for all t ∈ [0, T [,     3 1 8 d 1 3 2 2 ∞ ku1 (t)k3 + kω1 (t)k2 − C3 (1 + kΓ(t)χ{r≤R} kL∞ )kΓ(t)χ{r≤R} kL k∇|u1 (t)| 2 k22 + dt 3 2 9
kΓ0 kL∞ 1 1 + kΓ0 k2L∞ ku1 (t)k33 . + k∇ω1 (t)k22 ≤ C3 2 2 R (56) 12

8 . Since the real-valued 9 function y 7→ C3 (1 + y 2 )y is nondecreasing, then under the assumption that there exists R > 0 such that for any t ∈ [0, T [, kΓ(t)χ{r≤R} kL∞ ≤ γ0 , we get Let us introduce the unique constant γ0 > 0 satisfying C3 (1 + γ02 )γ0 =

C3 (1 + kΓ(t)χ{r≤R} k2L∞ )kΓ(t)χ{r≤R} kL∞ ≤

8 , 9

and from (56) we deduce that for all t ∈ [0, T [,  
d 1 1 kΓ0 kL∞ 3 2 ku1 (t)k3 + kω1 (t)k2 ≤ C3 1 + kΓ0 k2L∞ ku1 (t)k33 , 2 dt 3 2 R

which implies that for all t ∈ [0, T [,    
1 1 1 kΓ0 kL∞ d 1 3 2 2 3 2 ku1 (t)k3 + kω1 (t)k2 ≤ 3C3 1 + kΓ0 kL∞ ku1 (t)k3 + kω1 (t)k2 . dt 3 2 R2 3 2

(57)

(58)

Then thanks to Gronwall inequality, we deduce that for all t ∈ [0, T [,    
1 1 1 kΓ0 kL∞ 1 2 3 2 3 2 1 + kΓ0 kL∞ t , ku1 (t)k3 + kω1 (t)k2 ≤ ku1 (0)k3 + kω1 (0)k2 exp 3C3 3 2 3 2 R2

which concludes the proof.

Now, we finish with our main result. Theorem 5.1. Let u0 ∈ H m (R3 ) axisymmetric solenoidal vector field, with m ≥ 2 with Γ0 ∈ L2 (R3 )∩L∞ (R3 ). Let T > 0 be such that there exists u ∈ C([0, T [, H m (R3 ))∩L2 ([0, T [; H m+1 (R3 )) solution to the Navier-Stokes equations (2) for the initial data u0 . If u 6∈ C([0, T ], H m (R3 )) then we get, inf sup kΓ(t)χ{r≤R} kL∞ ≥ γ0 ,

R>0 t∈[0,T [

where γ0 > 0 is the absolute constant involved in Lemma 5.4. Proof. To get the proof, we assume first that inf sup kΓ(t)χ{r≤R} kL∞ < γ0 , then there exists R>0 t∈[0,T [

R > 0 such that sup kΓ(t)χ{r≤R} kL∞ ≤ γ0 .

t∈[0,T [

We derive first an estimate of ωθ ∈ L∞ L2 . Thanks to Lemma 5.4, we get that there exists a constant C > 0 such that for all t ∈ [0, T [,

3



!  

1 ωθ (t) 2

uθ (0) 3 1 ωθ (0) 2 u (t) 1 1 θ

+

≤

+

exp 3C kΓ0 kL∞ (1 + kΓ0 k2L∞ )T =: Q0 . 3 r 2 r 3 r 2 r R2 3

2

3

2

We multiply the first equation of (8) by ωθ and integrate it over t ∈ [0, T [, 1 d kωθ (t)k2L2 + 2 dt

Z

R3 .

(59) Then, we have for all

Z Z ωθ (t) 2 uθ (t) ur (t) 2 |∇ωθ (t)| + ωθ (t) − 2 ωr (t)ωθ (t). = r r r 2

13

(60)

On one hand, we have,

Z

ωθ (t) ur (t) 2

ωθ (t) ≤ kur (t)ωθ (t)k2

r r

2

ωθ (t)

≤ kur (t)k6 kωθ (t)k3 r

2

ωθ (t)

≤ ku(t)k6 kωθ (t)k2 kωθ (t)k6

r

2

1 ω (t) 3 3 θ

≤ C 2 k∇u(t)k22 k∇ωθ (t)k22

r 2

4 3

ω (t) 1 θ 2

≤ C 2 k∇u(t)k22

r + 4 k∇ωθ (t)kL2 , 2 1 2

1 2

(61)

where, we have used the Sobolev embedding H˙ 1 (R3 ) ֒→ L6 (R3 ) with C > 0 a constant and Young inequality. On the other hand, we have,

Z

uθ (t) uθ (t)

ωr (t)ωθ (t) ≤ 2kωθ (t)ωr (t)k 3 −2

2 r

r 3

uθ (t)

≤ 2kωθ (t)k6 kωr (t)k2 r 3 (62)

uθ (t)

≤ Ck∇ωθ (t)k2 k∇u(t)k2

r 3

2

1 u (t) θ 2

≤ C 2 k∇u(t)k22

r + 4 k∇ωθ (t)k2 . 3 Then, using (61) and (62), from (60), we deduce for all t ∈ [0, T [,



4

! Z Z

ωθ (t) 3 uθ (t) 2 ωθ (t) 2 1 1 d 2 2 2 2 ≤ C k∇u(t)k2

kωθ (t)k2 + |∇ωθ (t)| +

r + r , 2 dt 2 r 3 2 which implies that for all t ∈ [0, T [,

1d kωθ (t)k22 ≤ C 2 k∇u(t)k22 2 dt

4

!

ωθ (t) 3 uθ (t) 2



r + r 2 3

(63)

(64)

2 3

≤ C1 k∇u(t)k22 Q0 ,

where for the last inequality we have used (59) with C1 > 0 a constant. After an integration over [0, t] of inequality (64), we deduce that for all t ∈ [0, T [, Z t 2 2 2 3 k∇u(s)k22 ds kωθ (t)k2 ≤ kωθ (0)kL2 + 2C1 Q0 (65) 0 2 2 2 3 ≤ kωθ (0)kL2 + C1 Q0 ku0 k2 =: Ω0 , where we have used energy equality (4). Now, we multiply the second equation of (8) by ωr , the third equation of (8) by ωz , integrate them over R3 and sum the equations obtained, then we get for all t ∈ [0, T [, ! Z ωr (t) 2 1 d 2 2 2 2 |∇ωr (t)| + |∇ωz (t)| + (kωr (t)k2 + kωz (t)k2 ) + 2 dt r Z
∂r ur (t)ωr2 (t) + (∂z ur (t) + ∂r uz (t))ωr (t)ωz (t) + ∂z uz (t)ωz2 (t) . =

(66) Thanks to Lemma 2 in [6] and Theorem 3.1.1 in [3], we deduce that there exists a constant C2 > 0 such that for all t ∈ [0, T [, k∇ur (t)k2 ≤ C2 kωθ (t)k2 k∇uz (t)k2 ≤ C2 kωθ (t)k2 . 14

(67)

Furthermore, thanks to Cauchy-Schwarz inequality and Young inequality, we have for all t ∈ [0, T [, kωr (t)ωz (t)k2 ≤ kωr (t)k4 kωz (t)k4 (68) 1 1 ≤ kωr (t)k24 + kωz (t)k24 . 2 2 From (66), using Cauchy-Schwarz inequality, (67) and (68), we deduce that there exists a constant C3 > 0 such that for all t ∈ [0, T [, ! Z ωr (t) 2 1 d 2 2 2 2 |∇ωr (t)| + |∇ωz (t)| + (kωr (t)k2 + kωz (t)k2 ) + (69) 2 dt r ≤ C3 kωθ (t)k2 (kωr (t)k24 + kωz (t)k24 ).

Thanks to Interpolation inequality, Sobolev embedding H˙ 1 (R3 ) ֒→ L6 (R3 ), we deduce that there exists a constant C4 > 0 such for all t ∈ [0, T [, 3

1

kωr (t)k4 ≤ kωr (t)k24 kωr (t)k64 1

3

≤ C4 kωr (t)k24 k∇ωr (t)k24 , and also, 1

3

kωz (t)k4 ≤ C4 kωz (t)k24 k∇ωz (t)k24 . Then, from (69), we deduce that there exists a constant C5 > 0 such that for all t ∈ [0, T [, 2 ! Z ω (t) 1 d r |∇ωr (t)|2 + |∇ωz (t)|2 + (kωr (t)k22 + kωz (t)k22 ) + 2 dt r 1

3

≤ C5 kωθ (t)k2 kωr (t)k22 k∇ωr (t)k22 1

(70)

3

+C5 kωθ (t)k2 kωz (t)k22 k∇ωz (t)k22 .

Thanks to Young inequality, there exists a constant C6 > 0 such that for all t ∈ [0, T [, 1 3 1 C5 kωθ (t)k2 kωr (t)k22 k∇ωr (t)k22 ≤ C6 kωθ (t)k42 kωr (t)k22 + k∇ωr (t)k22 , 2

and also, 3 1 1 C5 kωθ (t)k2 kωz (t)k22 k∇ωz (t)k22 ≤ C6 kωθ (t)k42 kωz (t)k22 + k∇ωz (t)k22 . 2

Therefore, from (70), we deduce that for all t ∈ [0, T [,

! Z ωr (t) 2 1 d 1 2 2 2 2 (kωr (t)k2 + kωz (t)k2 ) + |∇ωr (t)| + |∇ωz (t)| + 2 dt 2 r ≤ C6 kωθ (t)k42 (kωr (t)k22 + kωz (t)k22 ),

(71)

which implies, 1 d (kωr (t)k22 + kωz (t)k22 ) ≤ C6 kωθ (t)k42 (kωr (t)k22 + kωz (t)k22 ). 2 dt Then, we integrate (72) over |0, t] and we obtain that for all t ∈ [0, T [, Z t kωθ (s)k42 (kωr (s)k22 + kωz (s)k22 ) ds. kωr (t)k22 + kωz (t)k22 ≤ kωr (0)k22 + kωz (0)k22 + 2C6

(72)

(73)

0

Then, thanks to (65) and (4), from (73), we deduce that for all t ∈ [0, T [, kωr (t)k22 + kωz (t)k22 ≤ kωr (0)k22 + kωz (0)k22 + 2C6 Ω20 ku0 k22 . 15

(74)

Then, thanks to (65) and (74), we deduce that lim sup kω(t)k2 < +∞. However since u 6∈ t→T

C([0, T ], H m (R3 )) with m ≥ 2, then (6) holds and we thus infer a contradiction with (6). Therefore we obtain that for any R > 0, sup kΓ(t)χ{r≤R} kL∞ ≥ γ0 ,

t∈[0,T [

which concludes the proof.

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