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A regularity criterion for the angular velocity component in the case of axisymmetric Navier– Stokes equations 1 ´ MILAN POKORNY Math. Institute of Charles University, Sokolovsk´a 83, 186 75 Praha 8, CZECH REPUBLIC e-mail: [email protected]

Introduction Let us consider the Navier–Stokes equations in the entire three-dimensional space, i.e. the system of PDE’s ) ∂u + u · ∇u − ν∆u + ∇p = 0 in (0, T ) × R3 ∂t (1) div u = 0 u(0, x) = u0 (x)

in R3 ,

where u : (0, T ) × R3 7→ R3 is the velocity field, p : (0, T ) × R3 7→ R is the pressure, 0 < T ≤ ∞, ν is the viscosity coefficient, u0 is the initial velocity and the forcing term is, for the sake of simplicity, considered to be zero. The question of smoothness and uniqueness of weak solutions to (1) is one of the most chalenging problems in the theory of PDE’s. It is known that if the weak solution belongs to1 Lt,s with 2t + 3s ≤ 1, s ∈ (3, +∞] and the input data are ”smooth enough” then the solution is smooth, see [8]. Instead of the smoothness of all velocity component we can require the additional information only for one velocity component, namely u1 ∈ Lt,s with 2t + 3s ≤ 21 , s > 6 (see [5]). In the case of the planar flow the weak solution is known to be unique and smooth as the data of the problem allow (see [4]). Thus a natural question, namely what can be said about the axisymmetric flow, appears.2 1

Key words and phrases: Axisymmetric flow, Navier–Stokes equations, regularity of systems of PDE’s. 1991 Mathematics Subject Classifications: 35Q35, 76D05. 1 By Lt,s we denote the anisotropic Lebesgue spaces Lt (I; Ls (R)). 2 Note that under an axisymmetric solution we understand a pair (u, p) such that in cylindrical coordinates (r, θ, z), r ∈ [0, ∞), θ ∈ [0, 2π) and z ∈ R, ur , uθ and uz , considered

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The first results in this direction were obtained in the late sixties for uθ = 0 (see [2], [9]) and recently also in [3]. The case uθ 6= 0, including the zaxis, was for the first time considered in the paper [6] where for ur ∈ Lr,s (QT ) 7 , with 2r + 3s ≤ 1, r ∈ [2, +∞], s ∈ (3, +∞] or for uθ ∈ Lr,s (QT ) with 2r + 3s ≤ 10 20 2 3 9 24 r ∈ [ 7 , +∞], s ∈ [6, +∞] and r + s ≤ 1 − 5s , r ∈ [10, +∞], s ∈ ( 5 , 6) the smoothness and thus also the uniqueness in the class of weak solutions satisfying the energy inequality was obtained. Note that the criterion on ur cannot probably be improved by the present technique while the regularity criteria on uθ are not optimal from the scaling argument. Here we would like to improve the regularity criteria on uθ in such a way that it (almost) undergoes the correct scaling, at least for certain range of s. Our main result is as follows THEOREM 1. Let u be a weak solution to problem (1) satisfying the energy inequality with u0 ∈ W 2,2 (R3 ) so that ∇u0 ∈ L1 (R3 ) and (u0 )θ r ∈ L∞ (R3 ). Let u0 be axisymmetric. Suppose further that the angular component uθ of u belongs to Lt,s for some t ∈ (2, +∞], s ∈ (4, +∞], 2t + 3s < 1. Then (u, p), where p is the corresponding pressure, is an axisymmetric strong solution to problem (1) which is unique in the class of all weak solutions satisfying the energy inequality.

Idea of the proof First note that our aim will be to get an estimate of ur in Lt,s with + 3s ≤ 1, s > 3. Then the smoothness of the solution follows from [6]. Because of the properties of the axisymmetric divergence-free functions (see [6]) it is actually enough to show that3 ωθ remains bounded in Ls1 ,t1 with 2 + s31 ≤ 2, 32 < s1 < 3. t1 We will also use the following inequalities: the Hardy type inequality

|ω |p  ω p  ω p2  2 

θ

θ

θ ≤ C(ε)

p+2−ε

+ ∇ r r p r 2 1 2 t

and the weighted estimate

u

ω

r

θ

1+q ≤ C(p, q) q r r p p in cylindrical coordinates, are independent of θ, and p, written in cylindrical coordinates, is also independent of θ. 3 i.e. the θ component of the vorticity

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which holds for 0 ≤ q < p2 , 1 < p < ∞. Both inequalities are shown in [7] where also the details of the proof presented below can be found. As already mentioned above we assume the right-hand side to be zero. It is well known that for sufficiently smooth data there exists a localin-time smooth solution to the three-dimensional Navier–Stokes solutions (see [1]). Moreover, if the data are axisymmetric then also the solution is axisymmetric. Denote by t∗ the supremum of the lenghts of the time intervals on which this solution exists and assume that t∗ < ∞. On subintervals of (0, t∗ ) we can work with smooth functions. Our aim will be to show the above mentioned estimated with a constant that does not blow up as t → t∗ . By standard continuation argument we thus obtain the contradiction to the definition of t∗ and thus t∗ = ∞. Let us also note that, after transforming the Navier–Stokes equations to the cylindrical coordinates, the continuity equation changes into ∂ur ur ∂uz + + =0 ∂r r ∂z while uθ and ωθ satisfy h 1 ∂ ∂u ∂uθ ∂uθ ∂uθ 1 ∂ 2 uθ uθ i θ + ur + uz + uθ ur − ν (r )+ − 2 =0 ∂t ∂r ∂z r r ∂r ∂r ∂z 2 r h   2 ∂ωθ ∂ωθ ∂ωθ ur 2 1 ∂ ∂ωθ ∂ ωθ ωθ i + ur + uz − ωθ + uθ ωr − ν r + − 2 = 0. ∂t ∂r ∂z r r r ∂r ∂r ∂z 2 r We first multiply the equation for ωθ by |ωθ |p−2 ωθ and by4 | ωrθ |q−2 ωrθ 1r and integrate over R3 (rdrdzdθ). We get

 d 1 1

ωθ q p kωθ kp + + dt p q r q Z  q p p 4(p − 1) |ωθ | 4(q − 1)  ωθ 2  2  2 2| + +ν + |∇|ω | ∇ θ r2 p2 q2 r Z Z Z ∂u ∂u |ur | 2 2 1 θ θ ≤ |ωθ |p + |uθ | |uθ | |ωθ |p−1 + |ωθ |q−1 q . r r ∂z r ∂z r We denote the terms on the right-hand side by I1 , I2 and I3 . To estimate I1 we use Z

 ω q  2  ω q  |ur |

θ

θ 2 p |ωθ | ≤ δ ∇ + C(δ)kωθ k22 + kωθ kpp r r r q 2 4

We must be careful here since some integrals may diverge; see [6] or [7] for more details.

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with q = 65 p, 65 ≤ p ≤ 2. The second term is “of lower order” and can be estimated quite easily. For the last term, we have for 0 < ε ≤ 1 (we use the Hardy type inequality) ! Z  Z  5 p  2 Z 5 p Z 5 5 p p  2 3 6 |u | ω |u | ω 12 6 θ θ θ θ + ∇ 5 p(1+ε) |I3 | ≤ C δ ∇ + + 5 p(1+ε)+2 r r 6 12 r r and thus, using the equation for uθ and the Gronwall inequality, we end up with 5

ω 5p  |uθ | 3 p 

θ 6 p kωθ kp + 5 + 5 p(1+ε) (t) r 6p r6 Z tZ 5 p |ωθ |p  ωθ 125 p  2  |uθ | 6 p  2 2 2 |∇(|ωθ | )| + 2 + ∇ +Cν + ∇ 5 p(1+ε) r r r 12 0 ! Z Z 5 5 t |uθ | 3 p |ur | |uθ | 3 p . + 5 p(1+ε)+2 ≤ C(u0 ) + C 5 r6 r 6 p(1+ε) r 0 Now, using the weighted estimate given above and the Young inequality we get Z 5

ω 5p

|u | 53 p |uθ | 3 p |ur |

θ 6

θ ≤ δ 5 + δ 5 p(1+ε)+2 5 p(1+ε) r r 2p 1 r6 r6 (2)  ω 56 p |u | 35 p  20pa

θ

θ +C(δ)kuθ k 10pa−5paε−18a+15p

5 + 5 p(1+ε) 10pa r 6p 1 6a−5p r6 with 1 < a < 2. Thus if uθ ∈ Lt,s with t=

20pa , 10pa − 5paε − 18a + 15p

s=

10pa 6a − 5p

(note that 2t + 3s = 1 − 2ε ) the last term can be estimated using the Gronwall that there exists p ≥ 23 such that inequality. As a < 2 we get for s > 20 3 10pa s = 6a−5p and the proof is finished. If 4 < s ≤ 20 we can only dispose with 3 the fact that the left-hand side of (2) is finite for some 56 < p < 32 . But 1 we multiply the equation for ωθ by |ωθ |− 2 ωθ and estimate all terms on the right-hand side using (2).5 The proof is finished. Unfortunately, the problem for 3 < s ≤ 4 remains open. 5

Note that, thanks to the Sobolev imbedding and interpolation, we get e.g. ωθ ∈ L4,2 ; 3 only the fact that ωθ is bounded in L∞, 2 would not be enough.

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Acknowledgement: The research was supported by the Grant Agency of the Czech Republic (grant No. 201/00/0768) and by the Council of the Czech Government (project No. 113200007).

References [1] Kiselev K.K., Ladyzhenskaya O.A.: On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid, Izv. Akad. Nauk SSSR 21 (1957) 655–680 (in Russian). [2] Ladyzhenskaya O.A.: On the unique global solvability of the Cauchy problem for the Navier–Stokes equations in the presence of the axial symmetry, Zap. Nauch. Sem. LOMI 7 (1968) 155–177 (in Russian). [3] Leonardi S., M´ alek J., Neˇcas J., Pokorn´ y M.: On axially symmetric flows in R3 , ZAA 18 (1999) 639–649. [4] Leray J.: Etude de diverses ´equations int´egrales non lin´eaires et de quelques probl`emes que pose l’hydrodynamique, J. Math. Pures Appl. IX. Ser. 12 (1933) 1–82. [5] Neustupa J., Novotn´ y, A., Penel, P.: An interior regularity criterion of a weak solution to the Navier–Stokes equations in dependence on one component of velocity, to appear in Topics in Mathematical Fluid Mechanics, a special issue of Quaderni di Matematica (2001). [6] Neustupa J., Pokorn´ y, M.: Axisymmetric flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Math. Boh. 126, No. 2 (2001). [7] Pokorn´ y, M.: A regularity criterion for the angular velocity component in the case of axisymmetric Navier–Stokes equations, submitted to the Proceedings of the 4th European congress on elliptic and parabolic problems, Rolduc 2001. [8] Serrin J.: The initial boundary value problem for the Navier–Stokes equations, In: Nonlinear Problems, ed. Langer R.E., University of Wisconsin Press (1963). [9] Uchovskii M.R., Yudovich B.I.: Axially symmetric flows of an ideal and viscous fluid, J. Appl. Math. Mech. 32 (1968) 52–61.