ЖЭТФ, 2020, том 158, вып. 2 (8), стр. 395-398

AN ANTISYMMETRIC SOLUTION OF THE 3D INCOMPRESSIBLE

NAVIER-STOKES EQUATIONS WITH

“TORNADO-LIKE” BEHAVIOR

C. Boldrighini^a, S. Frigio^b, P. Maponi^b, A. Pellegrinottic*, Ya. G. Sinaid**

^a Istituto Nazionale di Alta Matematica, Università di Roma Tre

00146, Rome, Italy

^b Scuola di Scienze e Tecnologie, Università di Camerino

62932, Camerino, Italy

^c Dipartimento di Matematica e Fisica, Università di Roma Tre

00146, Rome, Italy

^d Department of Mathematics, Princeton University

08544, Princeton NJ, USA

Received January 10, 2020,

revised version February 4, 2020

Accepted for publication February 4, 2020

Abstract. We consider a solution of the incompressible Navier-Stokes equations in R³, related to the singular

complex solutions of Li and Sinai [1], and such that a growth of the enstrophy S(t) is expected. The computer

simulations show that S(t) increases up to a time TE (singularities are excluded by axial symmetry). They also

reveal an interesting “tornado-like” behavior, with a sharp increase of speed and vorticity in an annular region,

as for some “extreme” weather phenomena.

DOI: 10.31857/S0044451020080167

The paper [1] is a first step in a plan to prove the

existence of a blow-up. The main idea is to apply dy-

1. Introduction. In recent times, results obtained

namical system techniques in order to control the trans-

by variants of the Navier-Stokes (NS) system indicate

fer of energy to the fine scales. The approach can also

that there are indeed smooth solutions that become

be applied to other models [7]. We give here a brief

singular (blow-up) at a finite time [2, 3]. As for evi-

description.

dence from computer simulations, the NS equations in

Consider the NS system in the whole space R³ with

3D are in general difficult to follow for high values of

no forcing, and viscosity ν

the velocity and the vorticity, and in absence of reli-

able theoretical guide-lines they are inconclusive (see,

∑

∂u

(x, t) +

u_j(x, t)

(x, t) =

e.g., [4]).

∂t

∂x_j

j=1

The singular solutions, if they exist, would describe

= νΔu(x,t) - ∇p(x,t), x = (x₁,x₂,x₃) ∈ R³,

(1)

sudden concentrations of energy in a finite region, as it

happens in tornadoes or hurricanes, for which no effec-

∑

∂u_j

tive model is now available. In fact, the main features

∇·u(x,t) =

(x, t) = 0, u(x, 0) = u₀(x), (2)

∂x_j

of the possible finite-time singularities (“blow-up”), are

the divergence of the total enstrophy [5], and the di-

where p is the pressure. Assuming ν = 1 and passing

vergence at some point of the absolute value of the

to the modified Fourier transform

velocity [6].

∫

v(k, t) =

u(x, t)e-i〈k,x〉dx,

(2π)³

(3)

^* E-mail: pellegri@mat.uniroma3.it

R³

^** Member of Russian Academy of Sciences

k = (k₁,k₂,k₃) ∈ R³,

395

C. Boldrighini, S. Frigio, P. Maponi et al.

ЖЭТФ, том 158, вып. 2 (8), 2020

where 〈·, ·〉 is the scalar product, the equation (1) can

The following Ansatz is formulated in [1]: for a

be written as a single integral equation:

class of Gaussian dominated initial data with support

as above, as p → ∞ the following asymptotics holds

∫^t

∏

v(k, t) = e-tk2 v₀(k) + e-(t-s)|k|2 ×

g(p)(Y, s) ∼ p(Λ(s))^p

g(Y_i)(H(Y) +

i=1

∫

〈v(k - k^′, s), k〉 P_kv(k^′, s) dk^′ds,

(4)

+ δ(p)(Y, s)), Y = (Y₁, Y₂, Y₃),

(7)

R³

√

where H is a fixed point, g(x) = exp(-x²/2)/

2π,

where P_kv = v -〈v,k〉|k|2 k is the solenoidal projector and

Λ is a strictly increasing smooth positive function and

v₀ the transform of u₀. In general v(k, t) is a complex

δ(p)(Y, s) → 0 as s → ∞. H is in fact a plane vector, as

function. Li and Sinai consider real solutions of (4),

its component along k⁽⁰⁾ vanishes by incompressibility.

which give in general complex solutions of (1), but if

The Ansatz (7) is proved in [1] for k⁽⁰⁾ = (0, 0, a),

v₀(k) is antisymmetric, the solution u(x, t) is also real

a > 0 and H⁽⁰⁾(Y) = c (Y₁,Y₂,0), with c > 0, and a

and antisymmetric in x.

finite-time blow-up is proved for a class of initial data

Multiplying v₀ by a constant A, which controls the

v₀. Both the total enstrophy and the total energy di-

initial energy, the solution of (4) is written as

verge as t ↑ τ (for complex function the energy equality

holds but it is not coercive).

v_A(k, t) = Ag⁽¹⁾(k, t)+

The object of our paper is the real flow which fol-

∫t

∑

lows from initial data obtained by antisymmetrizing the

+ e-k2(t-s) A^pg(p)(k, s)ds,

(5)

data associated to solutions that blow-up, namely

p=2

(

)

k²¹ + k²²

where

v₀(k) = k₁, k₂, -

g(k₁)g(k₂) ×

g⁽¹⁾(k, s) = e-sk2 v₀(k),

k₃

∫

[

]

g⁽²⁾(k, s) =

g⁽¹⁾(k - k^′, s), k P_kg⁽¹⁾(k^′, s)dk^′

× g(k₃-a)χ_b(|k-k⁽⁰⁾|)+g(k₃+a)χ_b(|k+k⁽⁰⁾|) ,

(8)

R³

and for p > 2

where a > b ≫ 1 and χ_b(r) is smooth with χ_b(r) = 0 if

r ≥ b, χ_b(r) = 1 if 0 ≤ r ≤ b-ϵ, for ϵ small enough. The

∫s

∫

∑

support of (8) is made of two regions around ±k⁽⁰⁾, and

g(p)(k, s) =

ds₁

ds₂ ×

the convolution g(p) is a sum of terms centered around

p1+p2=p

the points (0, 0, ℓa), ℓ = -p, . . . , p, with the main con-

p1,p2

∫

tribution coming for |ℓ| = O(√p). As the components

g(p1)(k - k^′, s₁), k P_kg(p2)(k^′, s₂)×

g(p) for large p are excited, the support moves quickly

R³

to the high k region, and an increase of the enstrophy

× e-(s-s1)(k-k′)²-(s-s2)(k′)²dk′ +

is expected.

Results of computer simulations. We

+ boundary terms.

(6)

used a special program for solutions of the integral

equation (4), created for the purpose of following the

The boundary terms involve g⁽¹⁾, and it can be shown

blow-up of the complex solutions, as described in [8],

that the series converges for small t.

where complex solutions of (4) could be followed up to

If, as in [1], the support of v₀ is concentrated in a

times close to the critical blow-up time. Our mesh in

sphere K_R of radius R centered around the point k⁽⁰⁾

k-space is a regular lattice centered at the origin with

with |k⁽⁰⁾| ≫ R, then g(p), which is a convolution, has

step δ = 1, with maximal configuration [-254, 254] ×

a support centered around pk⁽⁰⁾ with an effective di-

×[-254, 254]×[-3000, 3000]. We deal with about 5·10⁹

ameter of the order O(√p). By a standard rescaling

real numbers, close to the maximal capacity of modern

we write g(p)(Y, s) = g(p)(pk⁽⁰⁾ +

√p Y, s), and con-

supercomputers.

sider for large p the map g(p) → g(p+1). The possible

fixed points of that map control the excitation of the

Our main aim was to follow the behavior of the en-

high k-modes, i. e., of the fine structure components of

strophy and of the marginal distributions of the square

u(x, t).

vorticity in k-space, which describe the flow of energy

396

ЖЭТФ, том 158, вып. 2 (8), 2020

An antisymmetric solution of the 3D incompressible. . .

6. 10⁹

2.5

. 107

5. 10⁹

2.0

. 107

4. 10⁹

1.5

. 107

3. 10⁹

1.0

. 107

.109

5.0

. 106

1. 10

–300

-200

-100

100

200

300

-0.4

-0.2

0.2

0.4

Fig. 1. (Color online) Plots of the marginal distributions S3(k3, t) (a) and

S3(x3, t) (b) at the times t = 0 (blue), t = 400τ ≈ TV

(red), and t = 711τ ≈ TM (green)

max_x

3 |u(x,y)|

65000

60000

Velocity

65000

60000

55000

50000

45000

40000

35000

50000

30000

25000

20000

15000

10000

5000

1.7. 10^-12

100

200

300

400

500

600

700

Fig. 2. (Color online) Plot of the maximal velocity as a function of time (a), and the absolute value of the velocity field |u(x, t)|

on the plane x3 = 0.08 at the time t = 450τ (b)

to the microscale in physical space, and their behav-

the marginal densities of the enstrophy along the third∫

ior is stable with respect to refinements of the mesh.

axis: in k-space (a) S₃(k₃, t) =R2 |k|²|v(k, t)|²dk₁dk₂,

∫

An analysis comparing the accuracy of our program

and in x-space (b)

S₃(x₃, t) =R2 |ω(x, t)|²dx₁dx₂. On

with respect to that of finite-difference methods is un-

Fig. 1a we can see how the support moves into the high

der way.

|k|-region. As time grows the peaks of S₃(k₃, t) tend

We consider the initial data (8) with a = 30 and

to be close to the values k₃ ≈ ja, with j = ±1, ±2, . . .

multiplied by a constant A is such that the initial ener-

(green line), a modulated periodicity corresponding in

gy is E₀ = 62 · 10⁶. The study of the behavior of the

the plot of S₃(x₃) on Fig. 1b to two symmetric peaks

solutions as the parameters a and A vary is under way.

at x₃ = ±x₃ where x₃ grows in time but and as t ↑ T_E

As for the complex case [8] the large initial data en-

approaches a value x₃ ≈ π/a.

sure a short running time, which makes simulations

A remarkable fact is that the maximal value of the

possible. In what follows time is measured in units of

speed |u(x, t)| also grows, up to a time T_V

≈ 400τ

τ = 1.5625· 10-8.

(Fig. 2a). Moreover as we approach the time T_V the

high values of |u(x, t)| are concentrated in an annular

As expected, the total enstrophy S(t)

∫

region around a plane x₃ = x₃ as shown by Fig. 2b. The

|ω(x, t)|²dx, where ω(x, t) is the vorticity field,

R³

maximal values of the vorticity also grow, up to time

increases sharply up to a critical time T_E ≈ 711τ and

T_V and are concentrated in the same annular region.

then decays. In Fig. 1 we report the evolution in time of

397

C. Boldrighini, S. Frigio, P. Maponi et al.

ЖЭТФ, том 158, вып. 2 (8), 2020

3. Concluding remarks. The flow exhibits a

is partially supported by research funds of INdAM

“tornado-like” behavior, such as a rapid increase of the

(G.N.F.M.), M.U.R.S.T. and Università Roma Tre.

absolute values of velocity and vorticity in a confined

region, indicating that similar flows could provide a

The full text of this paper is published in the English

model for some class of such “extreme” phenomena.

version of JETP.

(Similar solutions probably exist also for compressible

fluids in a quasi-incompressible regime.)

A real blow-up is however excluded, due to the fact

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