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On the regularity of solutions of the 3D Axisymmetric
Navier-Stokes Equations with swirl
Léo Agélas
To cite this version:
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 three-dimensional (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, al-though 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 equa-tions in the axisymmetric case with general non-trivial swirl can not occur at the time T if the scale-invariant quantity inf
R>0t∈[0,T [sup kΓ(t)1{r≤R}kL
∞ is sufficiently small, where Γ = ruθ. To get
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))3and ∇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, ∞[; ˙H1(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,
∗
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−
t)−ǫ2 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
t∈[0,T [kΓ(t)1{r≤R}kL
∞ ≤ C1| ln R|−2 then 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 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 ∈
[0,12] such that sup
t∈[0,T [kΓ(t)1{r≤R}k
L∞ ≤ C1| ln R|− 3
2 then 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 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
R>0t∈[0,T [sup kΓ(t)1{r≤R}kL
∞ is
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[
and 0 < R ≤ Rε, k(Γ(t) − Γ(tǫ))1{r≤R}kL∞ ≤ ǫ 2 and by setting ˜Rǫ = ǫ 2(1 + kukL∞(R3×[0,t ǫ])) we get that for all t ∈ [0, tǫ] and for all 0 < R ≤ ˜Rǫ, kΓ(t)1{r≤R}kL∞ ≤ ˜Rǫku(t)kL∞ ≤
ǫ
2. Then by taking ¯Rǫ = min{Rǫ, ˜Rǫ}, we infer that for all t ∈ [0, T [, kΓ(t)1{r≤ ¯Rǫ}kL∞ ≤ ǫ and then we infer
that for any ǫ > 0, inf
R>0t∈[0,T [sup kΓ(t)1{r≤R}kL
∞ ≤ ǫ which means that inf
R>0t∈[0,T [sup kΓ(t)1{r≤R}kL
∞ =
0. Then, we conclude that the regularity criteria involved in [8, 9, 29, 30, 45] imply that inf
R>0t∈[0,T [sup kΓ(t)1{r≤R}kL
∞ = 0 which prove that their regularity criteria are more restrictive
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
t∈[0,T [kΓ(t)1{r≤R}k
L∞ ≤ γ
0 where γ0 >0 is an absolute constant.
Moreover, our criterion is bounded by kΓ0kL∞ thanks to (17), this feature eases the numerical
To obtain this result, we have been able to show the following energy estimate on [0, T [: d dt 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 + 8 9 − C(1 + kΓ(t)χ{r≤R}k 2 L∞)kΓ(t)χ{r≤R}kL∞ k∇|u1(t)| 3 2k2 2 +1 2k∇ω1(t)k 2 2 ≤ CkΓ0k L∞ R2 1 + kΓ0k 2 L∞ ku1(t)k33, (1) where u1= uθ r , ω1= ωθ
r and Γ = ruθ. Then, the paper is organized as follows:
• 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 Theorem5.1. 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 x 7−→ ϕ(x, t). For any vector x = (x
1, x2, x3) ∈ R3, we
denote by |x| the norm defined by |x| = v u u tX3
i=1
x2i. For any axisymmetric function f defined on
R3, for the sake of simplicity, the value f (x) with x = (x, y, z) ∈ R3 is denoted using coordinates cylindrical, f (r, z) with r = px2+ y2. For any d ≥ 1, Ω ⊂ Rd, we denote by C∞
0 (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 χ
Ω, 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) ∈ R+× R, χ{r≤R}(r, z) = 1 (resp
χ{r≥R}(r, z) = 1) for all r ≤ R and 0 elsewhere. The symbol Z
denotes the integral over R3
equal using cylindrical coordinates to Z ∞ 0 Z ∞ −∞ Z 2π 0
... r dθ dz dr. For any q > 1, the norm in Lq(R3) will be denoted by k · k
Lq 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
∂t + (u · ∇)u − ν∆u + ∇p = 0, ∇ · u = 0,
(2)
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,
Without loss of generality, in what follows, we assume that ν = 1.
Assuming u0 ∈ Hm(R3) for a given m ≥ 1, thanks to the results obtained in [34],
Theo-rem 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 [; Hm(R3))3∩
L2([0, T [; Hm+1(R3))3 to the Navier-Stokes Equations (2)-(3). Due to the regularity of solution of Navier-Stokes equation, u ∈ C([0, T [; Hm(R3))3 and thanks to the results obtained in [44],
[36], we get the energy equality, in other words, for all t ∈ [0, T [,
ku(t)k2L2(R3)3 + 2
Z t
0 k∇uk 2
L2(R3)3×3 = ku0k2L2(R3). (4)
Moreover if u 6∈ C([0, T ]; Hm(R3))3, then thanks to the results obtained in [34], Theorem 6.1
[11], Lemma 6.2 [12], we infer that, lim sup
t→T k∇u(t)kL
2(R3)3×3 = +∞, (5)
and thanks to Theorem 3.1.1 in [3], we have also, lim sup
t→T kω(t)kL
2(R3)3 = +∞, (6)
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 er = ( x r, y r,0), eθ = (− y r, x r,0), ez = (0, 0, 1) and r = p x2+ y2
In the above expression, uθ is called the swirl component of the velocity field u. For the
axisym-metric solutions, we can rewrite the equations (2) as follows : ∂uθ ∂t + ur ∂uθ ∂r + uz ∂uθ ∂z = L uθ− ur r uθ, ∂ur ∂t + ur ∂ur ∂r + uz ∂ur ∂z = L ur+ u2θ r + ∂rp, ∂uz ∂t + ur ∂uz ∂r + uz ∂uz ∂z = ∆uz+ ∂zp, ∂r(rur) + ∂z(ruz) = 0. (7)
For the axisymmetric vector field u, we can compute the vorticity ω = ∇ × u as follows, ω= ωrer+ ωθeθ+ ωzez,
where ωr= −(uθ)z, ωθ = (ur)z− (uz)r and ωz =1r(ruθ)r.
The operator L and ∆ is defined by : ∆ = ∂r2+1 r∂r+ ∂ 2 z, L = ∆ − r12 . (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 ∂t + ur ∂uθ ∂r + uz ∂uθ ∂z = L uθ− ur r uθ, ∂ωθ ∂t + ur ∂ωθ ∂r + uz ∂ωθ ∂z = L ωθ+ 1 r ∂u2θ ∂z + ur r ωθ, −L ψθ= ωθ (10)
where ur and uz can be expressed in terms of the angular component of the stream function ψθ
as follows : ur= − ∂ψθ ∂z , uz = 1 r ∂(rψθ) ∂r . (11)
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 Ck
s the axisymmetric divergence free subspace of Ck vector fields :
Csk(R3, R3) = {u ∈ Ck(R3, R3)| ∂θuz = ∂θur = ∂θuθ = 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 Ck
s(R+× R) is the function space defined by,
Csk(R+× R) = {f(r, z) ∈ Ck(R+× R)| ∂r2jf(0+, z) = 0, 0 ≤ 2j ≤ k}.
We can now define the Sobolev spaces for axisymmetric solenoidal vector fields : ˙
Hs1(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 · kHk(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 Navier-Stokes 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 ∈ Hk(R3)3 is an axisymmetric solenoidal vector field with k ≥ 1, then
there exists u0,θ∈ Hsk(R+× R), ψ0,θ ∈ ˙Hs1(R+× R) with L ψ0,θ ∈ Hsk−1(R+× R) such that u0 =
u0,θeθ+ ∇ × (ψ0,θeθ) and there exists a time T > 0 such that u = uθeθ+ ∇ × (ψθeθ) corresponds
to the unique strong solution to the Navier-Stokes equations (2) in the class C([0, T [; Hk(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)),
4
Estimates for axisymmetric solution
In this section, we recall some estimates on the quantity Γ = ruθ. For this, it is assumed that
u0 ∈ Hm 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 [; Hm(R3))∩L2([0, T [; Hm+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: ∂t+ b · ∇ − ∆ + 2 r∂r Γ = 0 (14)
with boundary conditions,
Γ|r=0= 0, (15)
with initial conditions,
Γ(x, 0) = Γ0(x) for a.e x ∈ R3, (16)
where, Γ0 = ru0,θ, b = urer+ uzez, b · ∇ = ur∂r+ uz∂z and div b = 0.
Note that in equation (14), the convection term has absorbed the term uruθ
r in the first equation (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)≤ kΓ0kLq(R3). (17)
5
Global regularity
In this section, we assume that u0∈ Hm is an axisymmetric solenoidal vector field, with m ≥ 2
and Γ0= ru0,θ∈ L2(R3) ∩ L∞(R3), then Theorem3.1holds 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 [, Hm(R3)) ∩ L2([0, T [; Hm+1(R3)) (see Section2). This section is devoted to the proof
of Theorem 5.1. The proof of our Theorem is obtained in three steps :
• First, thanks to the convection term, we eliminate an annoying term in (10), ur r ωθ, by using the change of unknowns from (uθ, ψθ, ωθ) to (u1, ψ1, ω1) (see (18)).
• Second, thanks to Lemmata 5.2and 5.3, we establish in Lemma5.4 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 [; H2(R3))3∩ L2([0, T [; H3(R3))3 and thanks to Lemmata 3-6 in [42], we deduce that,
u1 ∈ C([0, T [; H1(R3))
ω1 ∈ C([0, T [; L2(R3)) ∩ L2([0, T [; H1(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,
As in [21], from (10), we derive the following equivalent system for (u1, ω1, ψ1) : ∂tu1+ ur∂ru1+ uz∂zu1 = 2u1∂zψ1(t) + ∂r2u1+ ∂z2u1+ 3 r∂ru1 , ∂tω1+ ur∂rω1+ uz∂zω1 = ∂z(u1)2+ ∂r2ω1+ ∂z2ω1+ 3 r∂rω1 , − ∂r2ψ1+ ∂z2ψ1+ 3 r∂rψ1 = ω1 (21) where, ur = −r ∂ψ1 ∂z , uz = 1 r ∂(r2ψ 1) ∂r . (22)
Note that in the new system (21), the convection term has absorbed one of the vortex-stretching terms urωθ
r , which originally appears in the second equation of (10). In some sense, the con-vection term has already stabilized one of the potentially destabilized vortex-stretching terms in the above reformulation.
To obtain the proof of the crucial Lemma5.4, we use Lemma5.2 and Lemma 5.3. Lemma5.2
depends on Lemma 5.1which is an immediate consequence of CKN-type inequalities proved in [5].
Lemma 5.1. There exists a constant C > 0 such that for all v ∈ C0∞([0, +∞[\{0}) and α > 12, we have, Z ∞ 0 |v(r)| 2 r2(α−1) dr ≤ C Z ∞ 0 |v ′ (r)|2 r2α dr.
Here is the proof of Lemma5.2.
Lemma 5.2. There exists a constant C > 0 such that for all t ∈ [0, T [ and for all R > 0, we have, ku1(t)k44 ≤ C kΓ(t)χ{r≤R}kL∞k∇|u 1(t)| 3 2k2 2+ CkΓ0kL∞ R2 ku1(t)k 3 3.
Proof. Let R > 0. Consider the cut-off function ζ defined on R+ for which 0 ≤ ζ ≤ 1, ζ = 1
on [0,12], supp ζ ⊂ [0, 1]. Now, we consider the rescaled cut-off function ζR defined on R+ by
ζR(r) = ζ
r R
. For any x ∈ R3, we write x under the form x = (x′, z) where x′ ∈ R2. Then, we have, Z |u1(t)|4 = Z R3(|u1(x, t)|ζ R(|x′|) + |u1(x, t)|(1 − ζR(|x′|)))4 dx ≤ 4 Z R3( |u1(x, t)|ζ R(|x′|) )4 dx+ 4 Z R3( |u1(x, t)|(1 − ζ R(|x′|)) )4 dx ≤ 4 Z R3( |u1(x, t)|ζ R(|x′|) )4 dx+ 4 Z R3|u1(x, t)| 4χ {|x′ |≥R 2} dx.
With r = |x′|, we recall that Γ = ru
θ and Γ0 = ruθ(0), we notice that r2u1(t) = Γ(t), then
|u1(t)|4χ{r≥R 2}= |Γ(t)|χ{r≥R 2} r2 |u1(t)| 3
≤ 4|Γ(t)|R2 |u1(t)|3 and thanks to (17), we obtain,
|u1(t)|4χ{r≥R 2} ≤
4kΓ0kL∞
R2 |u1(t)| 3.
We consider u1(x, t) under the form u1(r, z, t), then we have, Z (|u1(x, t)|ζR(|x′|))4 = 2π Z ∞ −∞ Z ∞ 0 |u1(r, z, t)| 4ζ R(r)4r dr dz.
For a.e z ∈ R, thanks to Lemma 5.1used with v = (|u1(·, z, t)|ζR(r))2 and α = 32, we obtain,
Z ∞ 0 (|u1(r, z, t)|ζ R(r))4r dr ≤ C Z ∞ 0 |∂ r(|u1(r, z, t)|ζR(r))2|2r3dr = 4C Z ∞ 0 (|u1(r, z, t)|ζR (r))2|∂r(|u1(r, z, t)|ζR(r))|2r3dr ≤ 4C Z ∞ 0 |Γ(r, z, t)| |u1(r, z, t)| |∂ r(|u1(r, z, t)|ζR(r))|2r dr ≤ 8C Z ∞ 0 |Γ(r, z, t)| |u 1(r, z, t)|(|u1(r, z, t)|2ζR′ (r)2+ ζR(r)2|∂r|u1(r, z, t)||2) rdr ≤ 8C Z ∞ 0 |Γ(r, z, t)| |u1(r, z, t)|3kζ ′k2 L∞ R2 + 4 9χ{r≤R}|∂r|u1(r, z, t)| 3 2|2 r dr. (24) Thanks to Inequality (17), then from (24), we infer that there exists a constant C1 > 0 such
that, Z ∞ 0 (|u 1(r, z, t)|ζR(r))4r dr≤ C1kΓ0kL ∞ R2 Z ∞ 0 |u 1(r, z, t)|3rdr+ C1kΓ(t)χ{r≤R}kL∞ Z ∞ 0 |∂ r|u1(r, z, t)| 3 2|2rdr. Therefore, we obtain, Z R3(|u1(x, t)|ζR(|x ′|))4dx≤ C 1kΓ0kL ∞ R2 Z |u1(t)|3+ C1kΓ(t)χ{r≤R}kL∞ Z |∇|u1(t)| 3 2|2. (25)
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|2f ∈ L2(R2) and g ∈ H2(R2), we have, Z R2 f g ≤ C Z R2|x| 4f(x)2dx 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, kfkL2(R2) = √ 2πkζ(r)r12k L2(R +), k |x|2fk L2(R2) = √ 2πkζ(r)r52k L2(R +). (27)
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,
φK(r) = Z ∞ r 1 ρ Z ∞ ρ τ ζK(τ ) dτ dρ. (28)
Using successively the fact that supp ζK ⊂ [0, K] and |ζK| ≤ |ζ|, for all α ≥ 0 and for a.e τ > 0,
Using definition (28), inequality (29), Cauchy-Schwarz inequality and (27), we deduce that φK ∈ C1(]0, +∞[) and for all r > 0 and α > 12,
rα−12|φK(r)| ≤ K α+1 2 (α −12) p 2π(2α − 1)kfkL2(R2), rα+12|φ′ K(r)| ≤ Kα+12 p 2π(2α − 1)kfkL2(R2), rα|(rφ′ K(r))′| ≤ Kα+ 1 2|r 1 2ζ(r)|. (30)
Let us show that
r12φ
K ∈ L2([0, +∞[) and r
3 2φ′
K ∈ L2([0, +∞[). (31)
Using the first inequality of (30) with α = 3
4 and α = 2, we infer respectively that r
1
4|φK(r)| ≤
CKkfkL2(R2)and r 3
2|φK(r)| ≤ CKkfkL2(R2), where CK >0 is a real depending only on K. Then,
we get Z +∞ 0 rφK(r)2dr = Z 1 0 rφK(r)2dr+ Z +∞ 1 rφK(r)2dr = Z 1 0 r12(r 1 4φK(r))2dr+ Z +∞ 1 1 r2(r 3 2φK(r))2dr ≤ 53CK2kfk2L2(R2).
Therefore, we deduce that r12φK ∈ L2([0, +∞[). It remains to show that r 3 2φ′
K ∈ L2([0, +∞[).
Using the second inequality of (30) with α = 3
4 and α = 2, we infer respectively that r
5 4|φ′ K(r)| ≤ e CKkfkL2(R2)and r 5 2|φ′
K(r)| ≤ eCKkfkL2(R2), where eCK >0 is a real depending only on K. Then,
we get Z +∞ 0 r3|φ′K(r)|2dr = Z 1 0 r3|φ′K(r)|2dr+ Z +∞ 1 r3|φ′K(r)|2dr = Z 1 0 r12(r 5 4φ′ K(r))2dr+ Z +∞ 1 1 r2(r 5 2φ′ K(r))2dr ≤ 53CeK2kfk2L2(R2).
Therefore, we deduce that r32φ′
K ∈ L2([0, +∞[).
By using also the third inequality of (30) with α = 3
2 and thanks to (27), we infer, r32(rφ′
K)′∈ L2([0, +∞[). (32)
Then thanks to (31) and (32), by using twice Lemma5.1 with α = 3
2, we deduce, kr12φKk L2(R +) .kr 3 2φ′ KkL2(R +) = kr12(rφ′ K)kL2(R +) .kr32(rφ′ K)′kL2(R +) = kr52∆φ˜ K(r)k L2(R +), (33)
where for all r > 0, ˜∆φK(r) :=
Then, from (33), using (34), we obtain, kr12φKk L2(R +) .kr 5 2ζK(r)k L2(R +) ≤ kr52ζ(r)k L2(R +). (35)
We introduce the radial function ΦK defined on R2 by,
ΦK(x) = φK(|x|). (36)
Then, we get ∆ΦK(x) = ˜∆φK(|x|) and thanks to (34), we have
∆ΦK(x) = ζK(|x|) = ζ(|x|)χ{|x|≤K}= f (x)χ{|x|≤K}. Since, we have, Z R2 f g = Z R2 f(x)χ{|x|≤K}g(x) dx + Z R2 f(x)χ{|x|>K}g(x) dx ≤ Z R2 f(x)χ{|x|≤K}g(x) dx + Z R2 f(x)χ{|x|>K}g(x) dx . Then, we deduce, Z R2 f g ≤ Z R2 ∆ΦK(x)g(x) dx + Z R2 f(x)χ{|x|>K}g(x) dx . (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 R2 ∆ΦK(x)g(x) dx = Z R2 ΦK(x)∆g(x) dx ≤ kΦKkL2(R2)k∆gkL2(R2). (38)
Using the change of variables with polar coordinates, from (36), we observe, kΦKkL2(R2)= √ 2πkφK(r)r 1 2k L2(R
+), then thanks to (35) and (27), we deduce,
kΦKkL2(R2).k |x|2fkL2(R2). (39)
Then, using (39), from (38), we deduce, Z R2 ∆ΦK(x)g(x) dx . k |x|2fkL2(R2)k∆gkL2(R2). (40)
For the second term at the right hand side of inequality (37), thanks to Cauchy-Schwarz in-equality, we obtain, Z R2 f(x)χ{|x|>K}g(x) dx ≤ kf kL2({x∈R2,|x|>K})kgkL2(R2). (41)
Using (40) and (41), from (37), we obtain, Z R2 f g . k |x|2fkL2(R2)k∆gkL2(R2)+ kfkL2({x∈R2,|x|>K})kgkL2(R2). (42)
Since f ∈ L2(R2), then kfkL2({x∈R2,|x|>K}) → 0 as K → ∞. Then, taking the limit in inequality
(42) as K → ∞, we obtain, Z R2 f g . k |x|2fkL2(R2)k∆gkL2(R2),
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
t∈[0,T [kΓ(t)χ{r≤R}kL
∞ ≤ γ0,
then we get that for all t ∈ [0, T [, 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 ≤ 1 3ku1(0)k 3 3+ 1 2kω1(0)k 2 2 exp 3CkΓ0kL∞ R2 1 + kΓ0k 2 L∞ t .
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 [,
1 3 d dtku1(t)k 3 3+ 8 9 Z |∇|u1(t)| 3 2|2+2 3 Z ∞ −∞|u1(0, z, t)| 3dz = 2Z |u1(t)|3∂zψ1(t). (43)
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 |u1(t)|3∂zψ1(t) = Z R Z R2|u1 (x′, z, t)|3∂zψ1(x′, z, t)dx′ dz. (44)
Thanks to (19), (20) and Lemma5.3, there exists a constant C0 >0 such that for a.e z ∈ R,
Z
R2|u1
(x′, z, t)|3∂zψ1(x′, z, t)dx′ ≤ C0k |x′|2|u1|3(·, z, t)kL2(R2) k∇2x′(∂zψ1)(·, z, t)kL2(R2). (45)
From (44), thanks to (45) and Cauchy-Schwarz inequality, we get, Z |u1(t)|3∂zψ1(t) ≤ C0 Z Rk |x ′|2|u 1|3(·, z, t)k2L2(R2) dz 1 2 Z Rk∇ 2 x′(∂zψ1)(·, z, t)k2L2(R2) dz 1 2 = C0k |x′|2|u1(t)|3kL2(R3)k∇2 x′∂zψ1(t)kL2(R3). (46) Thanks to Lemma 1 in [22] used with u = ∂zψ1(t), f = ∂zω1(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
|∇2∂zψ1(t)|2 ≤ C1
Z
|∂zω1(t)|2. (47)
Then, thanks to (46) and (47), we deduce that there exists a real C2 > 0 such that for all
t∈ [0, T [, 2 Z |u1(t)|3∂zψ1(t) ≤ C2k |x′|2|u1(t)|3kL2(R3) Z |∂zω1(t)|2 1 2 . (48)
Recalling Γ = ruθ, with r = |x′|, we notice that |x′|2u1(t) = Γ(t), then |x′|2|u1(t)|3= |Γ(t)| |u1(t)|2,
then from (48), we obtain,
Further, we have k |Γ(t)| |u1(t)|2k22 = Z R3|Γ(x, t)| 2 |u 1(x, t)|4dx = Z R3|Γ(x, t)χ{|x ′|≤R}|2 |u1(x, t)|4dx+ Z R3|Γ(x, t)χ{|x ′|>R}|2 |u1(x, t)|4dx ≤ Z R3|Γ(x, t)χ{|x ′ |≤R}|2 |u1(x, t)|4dx+kΓ0k 3 L∞ R2 Z R3|u1(x, t)| 3dx,
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)| |x′|2 |u1(x, t)| 3 ≤ kΓ(t)k 3 L∞ R2 |u1(x, t)| 3 ≤ kΓ0k 3 L∞ R2 |u1(x, t)| 3 ( thanks to (17)).
Then, from (49), we obtain
2 Z |u1(t)|3∂zψ1(t) ≤ C22kΓ(t)χ{r≤R}k2L∞ku1(t)k4 4+ C22kΓ 0k3L∞ R2 ku1(t)k 3 3+ 1 4k∂zω1(t)k 2 2. (50) Using (50), from (43), we deduce that for all t ∈ [0, T [,
1 3 d dtku1(t)k 3 3+ 8 9k∇|u1(t)| 3 2k2 2 ≤ 1 4k∂zω1(t)k 2 2+C22kΓ(t)χ{r≤R}k2L∞ku1(t)k44+C22kΓ 0k3L∞ R2 ku1(t)k 3 3. (51) We multiply the first equation of (21) by ω1(t), integrate it over R3, use the incompressibility
condition (12), then we obtain for all t ∈ [0, T [,
1 2 d dtkω1(t)k 2 2+ Z |∇ω1(t)|2+ Z ∞ −∞|ω 1(0, z, t)|2dz = Z ω1(t)∂z(u1(t)2). (52)
By using integration by parts, Cauchy-Schwarz inequality and Young inequality, we deduce that for all t ∈ [0, T [, Z ω1(t)∂z(u1(t)2) = − Z ∂zω1(t)u1(t)2 ≤ k∂zω1(t)k2ku1(t)k24 ≤ 14k∂zω1(t)k22+ ku1(t)k44. (53)
Using (53), from (52), we obtain for all t ∈ [0, T [,
1 2 d dtkω1(t)k 2 2+ 3 4k∇ω1(t)k 2 2 ≤ ku1(t)k44. (54)
We sum inequalities (51) and (54), then, we obtain for all t ∈ [0, T [,
d dt 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 + 8 9k∇|u1(t)| 3 2k2 2+ 1 2k∇ω1(t)| 2 2 ≤ (1 + C22kΓ(t)χ{r≤R}k2L∞)ku1(t)k44 +C22kΓ0k 3 L∞ R2 ku1(t)k 3 3. (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 [,
Let us introduce the unique constant γ0 >0 satisfying C3(1 + γ02)γ0 =
8
9. Since the real-valued function y 7→ C3(1 + y2)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
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 dt 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 ≤ C3kΓ0kL ∞ R2 1 + kΓ0k 2 L∞ ku1(t)k33, (57)
which implies that for all t ∈ [0, T [, d dt 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 ≤ 3C3kΓ0kL ∞ R2 1 + kΓ0k 2 L∞ 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2 . (58)
Then thanks to Gronwall inequality, we deduce that for all t ∈ [0, T [, 1 3ku1(t)k 3 3+ 1 2kω1(t)k 2 2≤ 1 3ku1(0)k 3 3+ 1 2kω1(0)k 2 2 exp 3C3kΓ0kL ∞ R2 1 + kΓ0k 2 L∞ t ,
which concludes the proof.
Now, we finish with our main result.
Theorem 5.1. Let u0 ∈ Hm(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 [, Hm(R3))∩L2([0, T [; Hm+1(R3))
solution to the Navier-Stokes equations (2) for the initial data u0. If u 6∈ C([0, T ], Hm(R3)) then
we get,
inf
R>0t∈[0,T [sup kΓ(t)χ{r≤R}kL
∞ ≥ γ
0,
where γ0>0 is the absolute constant involved in Lemma 5.4.
Proof. To get the proof, we assume first that inf
R>0t∈[0,T [sup kΓ(t)χ{r≤R}kL
∞ < γ0, then there exists
R >0 such that
sup
t∈[0,T [kΓ(t)χ{r≤R}k
L∞ ≤ γ0.
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 [,
1 3 uθ(t) r 3 3 +1 2 ωθ(t) r 2 2 ≤ 1 3 uθ(0) r 3 3 + 1 2 ωθ(0) r 2 2 ! exp 3CkΓ0kL∞ R2 (1 + kΓ0k 2 L∞)T =: Q0. (59) We multiply the first equation of (8) by ωθ and integrate it over R3. Then, we have for all
On one hand, we have, Z u r(t) r ω 2 θ(t) ≤ kur(t)ωθ(t)k2 ωθ(t) r 2 ≤ kur(t)k6kωθ(t)k3 ωθ(t) r 2 ≤ ku(t)k6kωθ(t)k 1 2 2kωθ(t)k 1 2 6 ωθ(t) r 2 ≤ C32k∇u(t)k 3 2 2k∇ωθ(t)k 1 2 2 ωθ(t) r 2 ≤ C2k∇u(t)k22 ωθ(t) r 4 3 2 +1 4k∇ωθ(t)k 2 L2, (61)
where, we have used the Sobolev embedding ˙H1(R3) ֒→ L6(R3) with C > 0 a constant and
Young inequality. On the other hand, we have,
−2 Z u θ(t) r ωr(t)ωθ(t) ≤ 2kωθ(t)ωr(t)k32 uθ(t) r 3 ≤ 2kωθ(t)k6kωr(t)k2 uθ(t) r 3 ≤ Ck∇ωθ(t)k2k∇u(t)k2 uθ(t) r 3 ≤ C2k∇u(t)k22 uθ(t) r 2 3 +1 4k∇ωθ(t)k 2 2. (62)
Then, using (61) and (62), from (60), we deduce for all t ∈ [0, T [,
1 2 d dtkωθ(t)k 2 2+ 1 2 Z |∇ωθ(t)|2+ Z ωθ(t) r 2 ≤ C2k∇u(t)k22 ωθ(t) r 4 3 2 + uθ(t) r 2 3 ! , (63)
which implies that for all t ∈ [0, T [, 1 2 d dtkωθ(t)k 2 2 ≤ C2k∇u(t)k22 ωθ(t) r 4 3 2 + uθ(t) r 2 3 ! ≤ C1k∇u(t)k22Q 2 3 0, (64)
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 [,
kωθ(t)k22 ≤ kωθ(0)k2L2 + 2C1Q 2 3 0 Z t 0 k∇u(s)k 2 2ds ≤ kωθ(0)k2L2 + C1Q 2 3 0ku0k22 =: Ω0, (65)
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 [, 1 2 d dt(kωr(t)k 2 2+ kωz(t)k22) + Z |∇ωr(t)|2+ |∇ωz(t)|2+ ωr(t) r 2! = Z ∂rur(t)ωr2(t) + (∂zur(t) + ∂ruz(t))ωr(t)ωz(t) + ∂zuz(t)ω2z(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 ≤ C2kωθ(t)k2
Furthermore, thanks to Cauchy-Schwarz inequality and Young inequality, we have for all t ∈ [0, T [, kωr(t)ωz(t)k2 ≤ kωr(t)k4kωz(t)k4 ≤ 12kωr(t)k24+ 1 2kωz(t)k 2 4. (68)
From (66), using Cauchy-Schwarz inequality, (67) and (68), we deduce that there exists a con-stant C3 >0 such that for all t ∈ [0, T [,
1 2 d dt(kωr(t)k 2 2+ kωz(t)k22) + Z |∇ωr(t)|2+ |∇ωz(t)|2+ ωr(t) r 2! ≤ C3kωθ(t)k2(kωr(t)k24+ kωz(t)k24). (69)
Thanks to Interpolation inequality, Sobolev embedding ˙H1(R3) ֒→ L6(R3), we deduce that there
exists a constant C4 >0 such for all t ∈ [0, T [,
kωr(t)k4 ≤ kωr(t)k 1 4 2kωr(t)k 3 4 6 ≤ C4kωr(t)k 1 4 2k∇ωr(t)k 3 4 2, and also, kωz(t)k4 ≤ C4kωz(t)k 1 4 2k∇ωz(t)k 3 4 2.
Then, from (69), we deduce that there exists a constant C5 >0 such that for all t ∈ [0, T [,
1 2 d dt(kωr(t)k 2 2+ kωz(t)k22) + Z |∇ωr(t)|2+ |∇ωz(t)|2+ ωr(t) r 2! ≤ C5kωθ(t)k2kωr(t)k 1 2 2k∇ωr(t)k 3 2 2 +C5kωθ(t)k2kωz(t)k 1 2 2k∇ωz(t)k 3 2 2. (70)
Thanks to Young inequality, there exists a constant C6>0 such that for all t ∈ [0, T [,
C5kωθ(t)k2kωr(t)k 1 2 2k∇ωr(t)k 3 2 2 ≤ C6kωθ(t)k42kωr(t)k22+ 1 2k∇ωr(t)k 2 2, and also, C5kωθ(t)k2kωz(t)k 1 2 2k∇ωz(t)k 3 2 2 ≤ C6kωθ(t)k42kωz(t)k22+ 1 2k∇ωz(t)k 2 2.
Therefore, from (70), we deduce that for all t ∈ [0, T [,
1 2 d dt(kωr(t)k 2 2+ kωz(t)k22) + 1 2 Z |∇ωr(t)|2+ |∇ωz(t)|2+ ωr(t) r 2! ≤ C6kωθ(t)k42(kωr(t)k22+ kωz(t)k22), (71) which implies, 1 2 d dt(kωr(t)k 2 2+ kωz(t)k22) ≤ C6kωθ(t)k24(kωr(t)k22+ kωz(t)k22). (72)
Then, we integrate (72) over |0, t] and we obtain that for all t ∈ [0, T [,
kωr(t)k22+ kωz(t)k22≤ kωr(0)k22 + kωz(0)k22+ 2C6
Z t
0 kωθ(s)k 4
2(kωr(s)k22+ kωz(s)k22) ds. (73)
Then, thanks to (65) and (4), from (73), we deduce that for all t ∈ [0, T [,
Then, thanks to (65) and (74), we deduce that lim sup
t→T kω(t)k2
< +∞. However since u 6∈ C([0, T ], Hm(R3)) with m ≥ 2, then (6) holds and we thus infer a contradiction with (6).
Therefore we obtain that for any R > 0, sup
t∈[0,T [kΓ(t)χ{r≤R}k
L∞ ≥ γ
0,
which concludes the proof.
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