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Rough dependence upon initial data exemplified by explicit solutions and the effect of

viscosity

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2017 Nonlinearity 30 1097

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1097

Nonlinearity

Rough dependence upon initial data exemplified by explicit solutions and the effect of viscosity

Y Charles Li1

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States of America

E-mail: liyan@missouri.edu

Received 21 June 2015, revised 6 October 2016Accepted for publication 16 January 2017Published 3 February 2017

Recommended by Professor Edriss S Titi

AbstractIn this article, we present some interesting non-steady explicit solutions to the 2D Euler and Navier–Stokes equations. Explicit calculations on the explicit solutions show that Navier–Stokes (and Euler) equations have the novel property of rough dependence upon initial data in contrast to the sensitive dependence upon initial data found in chaos. Through the explicit calculations, we are able to obtain a lower bound on the norm of the Fréchet derivative of the solution operator at the explicit solutions to the Navier–Stokes equations. The lower bound approaches infinity as the Reynolds number approaches infinity. For Euler equations, this lower bound is indeed infinity. The rough dependence property in the inviscid case is closely related to the theorem of Cauchy. The viscous effect on the theorem of Cauchy and the rough dependence property is also studied.

Keywords: rough dependence on initial data, Euler equations, Navier–Stokes equationsMathematics Subject Classification numbers: Primary 76, 35; Secondary 34

1. Introduction

It is well-known that both Navier–Stokes and Euler equations are locally well-posed in the

Sobolev space Hs ( > +s 1d

2) where d is the spatial dimension [4, 5]. In two dimensions

(d = 2), the well-posedness is also global. In this article, we will focus on s = 3, then the

Y Charles Li

Explicit solutions

Printed in the UK

1097

NONLE5

© 2017 IOP Publishing Ltd & London Mathematical Society

30

Nonlinearity

NON

10.1088/1361-6544/aa59b8

Paper

3

1097

1108

Nonlinearity

London Mathematical Society

IOP

1 http://faculty.missouri.edu/~liyan

2017

1361-6544

1361-6544/17/031097+12$33.00 © 2017 IOP Publishing Ltd & London Mathematical Society Printed in the UK

Nonlinearity 30 (2017) 1097–1108 doi:10.1088/1361-6544/aa59b8

1098

solutions are in the space ([ ) )C T H0, ,0 3 , where T is either finite or infinite. The solution opera-tor Ft is a one-parameter family of maps on H3. For each fixed t, Ft maps the initial condition u(0) to the solution’s value at time t, u(t). F t is continuous in u(0). In [1], a simple example was presented showing that the solution operator Ft is not uniformly continuous in u(0) under the Euler dynamics. In [2, 3], the nowhere differentiability of Ft under the Euler dynamics was proved. In [7], we derived the upper bound on the Fréchet derivative of the solution opera-tor under the Navier–Stokes dynamics: ∥ ∥ ⩽∇ α α+F et t tRe 1 where α and α1 are constants depending only on the norm of the base solution. In this article, we present explicit solutions to both 2D Euler and 2D Navier–Stokes equations. These solutions show that under the Euler dynamics, ∇ = ∞Ft∥ ∥ , and under the Navier–Stokes dynamics, ∥ ∥ →∇ ∞Ft as the Reynolds number approaches infinity. We name the above nature of dependence of the solution operator on initial data as rough dependence on initial data.

2. Explicit solutions with rough dependence on initial data

In this section, we present explicit solutions to the 2D Euler equations and 2D Navier–Stokes equations, which show rough dependence on initial data. Consider the following 2D Euler equations

∂ + ⋅ ∇ = −∇ ∇ ⋅ =u u u p u, 0,t (2.1)

and 2D Navier–Stokes equations

∂ + ⋅ ∇ = −∇ + ∆ ∇ ⋅ =u u u p u u1

Re, 0,t (2.2)

under periodic boundary condition with period domain [ π0, 2 ] × [ π0, 2 ], where ( )=u u u,1 2 is the velocity, p is the pressure, and the spatial coordinate is denoted by ( )=x x x,1 2 . First we study the following simple explicit solution to the 2D Euler equations,

[ ( )] ∑ σ σ= − =γ=

∞

+un

n x t u1

sin , ,n

11

3 2 2 (2.3)

where ⩽γ< 11

2, and σ is a real parameter. We view this solution as a solution in the space

([ ) )∞C H0, ,0 3 , where H3 is the Sobolev space on the period domain [ π0, 2 ] × [ π0, 2 ].

We select the Sobolev space H3 because it is the largest Hs with integer s where the well- posedness is established for both the 2D Euler equations and the 2D Navier–Stokes equa-tions are well-posed in H3. The same construction in this article can be easily extended to any Sobolev space Hs ( R∈s ). Denote by Ft the solution operator of either the 2D Euler equa-tions or the 2D Navier–Stokes equations. For any fixed t, one can view Ft as a map in H3,

( ) → ( )F u u t: 0 .t

The initial condition of the solution (2.3) is

( ) ∑ σ= =γ=

∞

+un

nx u1

sin , .n

11

3 2 2

By varying σ, we get a variational direction of the initial condition,

( ) ( ) σ= =u ud 0 0, d 0 d ,1 2

which leads to the directional derivative ∂σFt of Ft:

Y Charles Li Nonlinearity 30 (2017) 1097

1099

[ ( )] ∑σσ

σ∂∂=

−−

∂∂=γ

=

∞

+u t

nn x t

ucos , 1.

n

1

12 2

2

The norm of ∂σFt is given by

∑

∑

π π

π π

∂ = + + + +

= + + + +

= ∞ >

σ γ

γ γ γγ

=

∞

+

=

∞

+ +−

⎛

⎝⎜

⎞

⎠⎟

F t n n nn

tn n n

n

t

4 2 11

4 21 1 1

, 0 ,

tH

n

n

2 2 2 2

1

2 4 64 2

2 2 2

14 2 2 2 2

2 2

3∥ ∥ ( )

( )

(2.4)

where the last series is divergent when ⩽γ< 11

2. Thus the directional derivative ∂σFt does

not exist. Therefore the derivative ∇Ft does not exist in view of the fact that the norm of the derivative ∇Ft is greater than or equal to the norm of the directional derivative ∂σFt. In fact, the solution operator Ft is nowhere differentiable [2].

The corresponding solution of (2.3) to the 2D Navier–Stokes equations (2.2) is

[ ( )] ∑ σ σ= − =γ=

∞

+−u

nn x t u

1e sin , ,

n

n t1

13

Re 2 2

2

(2.5)

which is also a solution in the space ([ ) )∞C H0, ,0 3 . The directional derivative ∂σFt of Ft is

[ ( )] ∑σσ

σ∂∂=

−−

∂∂=γ

=

∞

+−u t

nn x t

ue cos , 1.

n

n t1

12

Re 22

2

The norm of ∂σFt is given by

∑π π∂ = + + + +σ γ=

∞

+−F t n n n

n

ne4 2 1 .t

Hn

n t2 2 2 2

1

2 4 62

6 22

Re3

2

∥ ∥ ( )

Let

( )ξ ξ= ξ−g t e .t2 2

Re (2.6)

The maximum of ( )ξg is given by

( ) ⎜ ⎟⎛⎝

⎞⎠ξ ξ= − =′ ξ−g t

te 1

2

Re0,

t2 2Re

that is,

ξ =t

Re

2, (2.7)

where the maximal value of g is

= −g

te

Re

2.1

Thus

σ∂ +σ

⎛

⎝⎜

⎞

⎠⎟F t

eu

1Re

1

20 .t

H H3 3∥ ∥ ⩽ ∥ ( )∥

Y Charles Li Nonlinearity 30 (2017) 1097

1100

Notice that for the full derivative ∇Ft, the upper bound is given as [7],

∥ ∥ → ⩽∇ α α+F e ,tH H

t tRe3 3 1 (2.8)

where α and α1 are two constants depending on the H3 norm of the base solution. From (2.7), let

=⎡

⎣⎢

⎤

⎦⎥

⎛⎝⎜

⎞⎠⎟n

t t

Re

2, the integer part of

Re

2,

then

⩽ ⩾<t

nt t

1

2

Re

2

Re

2, when

Re

21.

We have

π

π

π π

π π

π π

π π

∂ >

+ + + +

> +

+

= +

+

σ

γ

γ

γ

γγ

γγ

+−

− −

−−

−

−

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

F

t n n nn

ne

t n e

tt

e

et

t

et

t

4

2 1

4 2

4 21

2

Re

2

22 Re

2 2

1

22

2 Re

2 2.

tH

n t

n t

2 2

2 2 2 4 62

6 22

Re

2 2 2 2 2 2Re

2 2 22 2

1

21 2

1 2

3

2

2

∥ ∥

( )

⩾

( )

⩾

Thus

∥ ∥⎛⎝⎜

⎞⎠⎟π

π∂ > +σ

γγ−

Fe

tt

2Re

2 2.t

H

1

3 (2.9)

As →∞Re ,

∥ ∥ →∂ ∞σF ,tH3

thus

∥ ∥ →∇ ∞F .tH3

From another perspective, one can also obtain a lower bound for ∥ ∥∂σFtH3. Fix a t > 0, for each

n, the maximum (2.7) specifies a Reynolds number ( )Re n (with ξ = n2),

( ) = tnRe 2 ,n 2

where

( ) = −g n e t n .2 1 2 2

Y Charles Li Nonlinearity 30 (2017) 1097

1101

Since∥ ∥

( )

π

π

π π

∂ >

+ + + +

> +

σ

γ

γ

+−

− −

F

t n n nn

ne

t n e

4

2 1

4 2 ,

tH

n t

n t

2 2

2 2 2 4 62

6 22

Re

2 2 2 2 2 2Re

3

2

2

we have

∥ ∥ ( )π∂ > + =σγ−F

t

en2 1

2, when Re Re .t

Hn

22 2

3

Therefore, as →( ) ∞Re n ( →∞n ),

∥ ∥ →∂ ∞σF .tH3

More general solutions with the same property of rough dependence on initial data can be derived as follows. We expand the vorticity into a Fourier series

/ Z∑ω ω=∈

⋅ek

kk x

0

i

2

where

( ) ( )ω ω= = =− k k k x x x, , , , .k k 1 2 1 2

When the velocity has zero spatial mean, the 2D Euler equations can be re-written as

( )∑ω ω ω== +

A p q˙ ,kk p q

p q

where

( ) ( )= | | − | |− −A p q q pp qp q,

1

2.2 2 1 1

2 2 (2.10)

When ∥p q,

( ) =A p q, 0.

This implies that for any / Z∈k 02 ,

/

( )

Z∑ω =∈

⋅C en

nn k x

0

i (2.11)

is a steady solution to the 2D Euler equation (in vorticity form) [6], where Cn’s are complex constants, and =−C Cn n. In terms of velocity, this solution has the form

/

( )

/

( )

Z

Z

∑

∑

=| |

= −| |

∈

⋅

∈

⋅

uk C

n k

uk C

n k

ie ,

ie .

n

n n k x

n

n n k x

10

22

i

20

12

i

Under a translation in velocity, this solution is transformed into the following time-dependent solution,

Y Charles Li Nonlinearity 30 (2017) 1097

1102

/

[ ( ) ( )]

/

[ ( ) ( )]

Z

Z

∑

∑

σ

σ

= +| |

= −| |

σ σ

σ σ

∈

− + −

∈

− + −

uk C

n k

uk C

n k

ie ,

ie ,

n

n n k x t k x t

n

n n k x t k x t

1 10

22

i

2 20

12

i

1 1 1 2 2 2

1 1 1 2 2 2

(2.12)

where σ1 and σ2 are real constants. By choosing

( ⩽ )γ= <γ+Cn

1,

1

21 ,n 2 (2.13)

we get a solution which has the same property of rough dependence on initial data as the solu-tion (2.3). The initial condition of the solution is

( )

( )

/

( )

/

( )

Z

Z

∑

∑

σ

σ

= +| |

= −| |

γ

γ

∈+

+

∈+

+

uk

n k

uk

n k

0i

e ,

0i

e .

n

n k x k x

n

n k x k x

1 10

23 2

i

2 20

13 2

i

1 1 2 2

1 1 2 2

By varying σ1 (or σ2), we get a variational direction

( ) ( )σ= =u ud 0 d , d 0 0,1 1 2

and the corresponding directional derivative ∂σ Ft1 of the solution operator Ft is

/

[ ( ) ( )]

/

[ ( ) ( )]

Z

Z

∑

∑

σ

σ

∂∂= +

| |

∂∂= −

| |

γσ σ

γσ σ

∈+

− + −

∈+

− + −

u t

n

k k

k

u t

n

k

k

1 e ,

e .

n

n k x t k x t

n

n k x t k x t

1

1 02

1 22

i

2

1 02

12

2i

1 1 1 2 2 2

1 1 1 2 2 2

The norm of ∂σ Ft1 is given by

∥ ∥ ( )/ ∑π π∂ = +

| |+ | | + | | + | | = ∞σ γ

∈+

ZF t

k

kn k n k n k

n4 4 1

1,t

Hn

2 2 2 12

20

2 2 4 4 6 64 21 3

(2.14)

when t > 0 and ≠k 01 .Under the viscous effect, the corresponding solution of (2.12) to the 2D Navier–Stokes

equations (2.2) is given by

/

[ ( ) ( )]

/

[ ( ) ( )]

∑

∑

σ

σ

= +| |

= −| |

σ σ

σ σ

∈

−| |

− + −

∈

−| |

− + −

Z

Z

uk C

n k

uk C

n k

ie e ,

ie e .

n

nn k

t n k x t k x t

n

nn k

t n k x t k x t

1 10

22

Re i

2 20

12

Re i

2 2

1 1 1 2 2 2

2 2

1 1 1 2 2 2

(2.15)

We still choose Cn as in (2.13). The directional derivative ∂σ Ft1 is given by

∑

∑

σ

σ

∂∂= +

| |

∂∂= −

| |

γσ σ

γσ σ

∈+

−| |

− + −

∈+

−| |

− + −

Z

Z

u t

n

k k

k

u t

n

k

k

1 e e ,

e e .

n

n kt n k x t k x t

n

n kt n k x t k x t

1

1 02

1 22

Re i

2

1 02

12

2Re i

2 2

1 1 1 2 2 2

2 2

1 1 1 2 2 2

/

[ ( ) ( )]

/

[ ( ) ( )]

Y Charles Li Nonlinearity 30 (2017) 1097

1103

The norm of ∂σ Ft1 is given by

∥ ∥

( )/ ∑π π

∂ =

+| |

+ | | + | | + | |

σ

γ∈

+−

| |

Z

F

tk

kn k n k n k

n

ne4 4 1 .

tH

n

n kt2 2 2 1

2

20

2 2 4 4 6 62

6 22

Re

1 3

2 2

By the result on the function ( )ξg (2.6) with ξ = | |n k2 2, we get

∥ ∥ ⩽ ∥ ( )∥σ σ

∂+

+| || |

σ

⎛

⎝⎜⎜

⎞

⎠⎟⎟F

k

kt

eu

1Re

1

20 .t

H H

12

22

11 3 3

From (2.7), let

=| | | |

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎛⎝⎜

⎞⎠⎟n

k t k t

1 Re

2, the integer part of

1 Re

2,

then

⩽ ⩾| |

<| | | |k t

nk t k t

1

2

1 Re

2

1 Re

2, when

1 Re

21.

We have

π

π

π π

π π

ππ

ππ

∂ >

+| |

+ | | + | | + | |

+ | |

+ | || |

= + | |

+ | |

σ

γ

γ

γ

γ γγ

γ γγ

+−

| |

− −| |

−−

+−

+−

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

F

tk

kn k n k n k

n

ne

t k k n e

t k kk t

e

ek k t

t

ek k t

t

4

4 1

4 4

4 41

2

1 Re

2

22 Re

2 2

1

22

2 Re

2 2.

tH

n kt

n kt

2 2

2 2 12

22 2 4 4 6 6

2

6 22

Re

2 2 212 4 2 2 2

Re

2 2 212 4

2 21

21

11 2

11

1 2

1 3

2 2

2 2

∥ ∥

( )

⩾

⩾

( )

⩾

Thus

∥ ∥⎛⎝⎜

⎞⎠⎟π

π∂ > + | |σ

γ γγ

+−

Fe

k k tt

22 Re

2 2.t

H 11

1

1 3 (2.16)

As →∞Re ,

∥ ∥ →∂ ∞σ F ,tH1 3

thus

∥ ∥ →∇ ∞F .tH3

Y Charles Li Nonlinearity 30 (2017) 1097

1104

Now we go back to the formula (2.10), when | | = | |p q ,

( ) =A p q, 0.

This implies that

∑ω =| |=

⋅C ce , is a constantk c

kk xi ( )

is a steady solution of the 2D Euler equations (in vorticity form) [6], where Ck’s are complex constants, and =−C Ck k . In terms of velocity, this solution has the form

∑

∑

=| |

= −| |

| |=

⋅

| |=

⋅

uk C

k

uk C

k

ie ,

ie .

k c

k k x

k c

k k x

12

2i

21

2i

Under a translation in velocity, this solution is transformed into the following time-dependent solution,

[ ( ) ( )]

[ ( ) ( )]

∑

∑

σ

σ

= +| |

= −| |

σ σ

σ σ

| |=

− + −

| |=

− + −

uk C

k

uk C

k

ie ,

ie ,

k c

k k x t k x t

k c

k k x t k x t

1 12

2i

2 21

2i

1 1 1 2 2 2

1 1 1 2 2 2

(2.17)

where σ1 and σ2 are real constants. A special case of (2.17) was given in [1]. By choosing

σ σσ

= =n

,1 2

where σ is a parameter and n is any positive integer, and

( ) ( ) ( ) ( )= − −

= − −− − − −

k n n n n

C n n n n

, 0 , , 0 , 0, , 0, ,

i1

2, i

1

2, i

1

2, i

1

2,k

s s s s1 1 1 1

where R∈s , one ends up with the sequence of solutions labeled by n given in [1]. Notice the interesting fact that this sequence does not converge to zero function in Hs, in fact, it does not have a limit in Hs, and it is a bounded sequence. By choosing two values of σ, e.g. ±1, one gets two sequences of solutions. At t = 0, the two sequences approach each other in Hs as →∞n , but when t > 0, the two sequences do not approach each other in Hs as →∞n . This shows that the solution operator is not uniformly continuous in Hs. On the other hand, this sequence of solutions does not show that the solution operator is not differentiable. In particular, in con-trast to (2.4) and (2.14), the norm of the corresponding directional derivative is finite for any of the solutions in this sequence (more generally in (2.17)) since the solution only contains a finite number of Fourier modes. The corresponding solution of (2.17) to the 2D Navier–Stokes equations is given by

[ ( ) ( )]

[ ( ) ( )]

∑

∑

σ

σ

= +| |

= −| |

σ σ

σ σ

| |=

− | | − + −

| |=

− | | − + −

uk C

k

uk C

k

ie e ,

ie e .

k c

k k t k x t k x t

k c

k k t k x t k x t

1 12

2

1Re i

2 21

2

1Re i

21 1 1 2 2 2

21 1 1 2 2 2

Y Charles Li Nonlinearity 30 (2017) 1097

1105

For such a solution, in contrast to (2.16), the norm of the corresponding directional derivative is uniformly bounded for all values of the Reynolds number, due to the fact that the solution contains only a finite number of Fourier modes.

3. Extension of a theorem of Cauchy to the viscous case

To gain a better understanding on the geometric intuition behind the rough dependence phenom enon, one needs to explore the theorem of Cauchy on vorticity. We start with the composition map

( )φ φ f f, , (3.1)

where ( )R∈f H3 , ( )R Rφ− ∈I H ,3 , I is the identity map, and φ is monotone. f can be vector-valued or complex-valued, but we restrict it to be real-valued for simplicity. φ is actually a diffeomorphism on R. It was shown in [2, 3] that this composition map is continuous but nowhere locally uniformly continuous and nowhere differentiable. The geometric intuition for this claim is as follows: for f, one chooses a sequence of sharp pulses with the same norm, but their small supports approach one point. Corresponding to each such f pulse, one choose two φ’s, one is the identity map, and the other is the identity map plus a perturbation map. The perturbation map is able to shift the small support of the f pulse completely out of its original location (i.e. the support of the pulse and the preimage of the support under the non-identity φ have no overlap). As a sequence, these perturbation maps approach the zero map. Now one ends up with two sequences of ( φf , ). The two sequences share the same f sequence of pulses, and the elements of the two φ sequences are given by the two φ maps just defined above corre-sponding to each f pulse (i.e. the elements of one φ sequence are always the identity map, and the elements of the other φ sequence are the identity map plus the perturbation maps). The two sequences of ( φf , ) approach each other since the perturbation map sequence approaches the zero map, but the images of the two sequences of ( φf , ) under the composition map (3.1) do not approach each other since the perturbation maps shift the supports of the f pulses com-pletely out of their original locations. This is the geometric intuition for the composition map (3.1) being nowhere locally uniformly continuous. Nowhere differentiability follows from nowhere uniform continuity via a relatively easy argument.

The theorem of Cauchy on vorticity enables the solution operator of the Euler equations to have the same property as the composition map (3.1). One can view the theorem of Cauchy (3.4) from the perspective of the composition map (3.1) as

( ( ) ( )) ( )ω ξ ωx x t x t, 0 , , , , (3.2)

where ω and ξ are defined in details below, ω is the vorticity, and ξ is the fluid particle trajec-tory map which maps the initial location of the fluid particle to its location at time t. In contrast to the composition map (3.1), ( )ξ x t, depends upon ( )ω x, 0 , but this does not pose a funda-mental obstacle for the basic idea introduced above for the composition map to go through as shown in [2]. One can again define two sequences of ( )ω x, 0 . Both contain the pulses, but one of them also contains the perturbations which generate ( )ξ x t, perturbation maps that can shift the supports of the pulses completely out of their original locations. The key is to properly separate the small supports for the pulses and the perturbations. The specific technical details of the arguments in [2] also involve the velocity variables. In fact, it was shown in [2] that the solution operator in the velocity variables of the Euler equations is nowhere locally uniformly continuous and nowhere differentiable.

The theorem of Cauchy (3.4) together with the fluid particle trajectory equation (3.3), Biot–Savart law and the incompressibility condition, is equivalent to the Euler equations.

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Therefore, the theorem of Cauchy (3.4) is a representation of the solution operator of the Euler equations. The composition map type nature of the theorem of Cauchy (3.4) leads to the nowhere differentiability of the solution operator of the Euler equations. On the other hand, the solution operator of the Navier–Stokes equations is everywhere differentiable. The viscos-ity clearly plays the key role here. The viscous effect erases the composition map type nature of the original theorem of Cauchy (3.4). Therefore, one should explore the viscous effect on the theorem of Cauchy (3.4).

We consider 3D fluid flows (2D results can be easily obtained from the 3D ones by 2D reduction). Let ( )ξ x t, be the fluid particle trajectory starting from x at the initial time t = 0,

( ) ( ( ) ) ( )ξ ξ ξ∂ = =x t u x t t x x, , , , , 0 ,t (3.3)

where ( )=u u u u, ,1 2 3 is the fluid particle velocity, ( )=x x x x, ,1 2 3 is the initial fluid particle location, and ( )ξ ξ ξ ξ= , ,1 2 3 is the fluid particle location at time t.

For any fixed t, one can think ( )ξ x t, as a map of the fluid domain. By the Liouville theorem [8] (page 48), ξ is a volume-preserving map in time, and any volume in the fluid domain is preserved during the fluid motion. The following is a theorem proved by Cauchy [8, 9].

Theorem 3.1. The vorticity ( )ω ω ω ω= , ,1 2 3 along a fluid particle trajectory under the Euler dynamics, evolves according to

( ( ) ) ( ) ( )ω ξ ξ ω=x t t x t x, , , , 0 ,i i j j, (3.4)

where i = 1, 2, 3, repeated index j means the usual summation convention, and ‘,j’ means spatial derivative in xj.

Under the 2D reduction,

( ( ) ) ( )ω ξ ω=x t t x, , , 0 ,

where the ω is actually the third component of vorticity.It is not easy to extend the equation (3.4) to the viscous case by following the argument

in [8] (pages 64–65). Nevertheless, an extension of (3.4) to the viscous case was established in [10] (page 154) with a formula involving ξ and the acceleration of the fluid particle. The formula (3.6) we obtain below involves ξ and vorticity. From the Eulerian perspective, it is easier to estimate the vorticity than the particle acceleration. Of course, the two formulas have to be equivalent.

Since vorticity is a pseudovector and in fact a tensor, the theorem of Cauchy has a tensor version [2]. As shown below, the tensor version can be easily extended to the viscous case. One can introduce the tensor version of vorticity as a ×3 3 matrix Ω with entries

Ω = −u u .ij i j j i, ,

Then the theorem of Cauchy takes the following form [2],

Theorem 3.2. The matrix vorticity along a fluid particle trajectory under the Euler dynam-ics, evolves according to

( ) ( )ξ ξ ξΩ = Ωt x, , 0 .m i mk k j ij, , (3.5)

Equation (3.5) is a consequence of the equation

( ( ) )ξ ξ ξ∂ Ω =t, 0.t m i mk k j, ,

The classical form of the Cauchy theorem (3.4) can be obtained from (3.5) when using the ω variable. Equation (3.5) can be extended to the viscous case rather easily.

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Theorem 3.3. The matrix vorticity along a fluid particle trajectory under the Navier–Stokes dynamics, evolves according to

( ( ) ) ( )ξ ξ ξ ξ ξ ξ∂ Ω = ∆Ωt t,1

Re, .t m i mk k j m i mk k j, , , , (3.6)

Proof. Starting from the Navier–Stokes equations

∂ + = − + ∆u u u p u1

Re,t i j i j i i, ,

we get

∂ Ω + Ω + Ω + Ω = ∆Ωu u u1

Re.t ij ik k j k i kj k ij k ij, , , (3.7)

From (3.3), we have

ξ ξ∂ = u .t i j i k k j, , , (3.8)

The material derivative of ( )ξΩ t,ij is given by

( )ξ∂ Ω = ∂ Ω + Ωt u, .t ij t ij k ij k, (3.9)

Now we can use (3.8) and (3.9) to calculate the material derivative

( ( ) )[ ]

[ ][ ]

ξ ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξ

∂ Ω = Ω

+ ∂ Ω + Ω + Ω

= Ω + ∂ Ω + Ω + Ω

= ∂ Ω + Ω + Ω + Ω

t u

u u

u u u

u u u

,

,

t m i mk k j l i m l mk k j

m i t mk l mk l k j m i mk k l l j

m i l m lk k j m i t mk l mk l k j m i ml l k k j

m i t mk l mk l ml l k l m lk k j

, , , , ,

, , , , , ,

, , , , , , , , ,

, , , , ,

by rearranging the dummy indices. Using (3.7), we get (3.6).

Now we discuss the key role played by the viscosity on the rough dependence property. In the inviscid case, the relation (3.5) describes the composition map type nature:

( ( ) ( )) ( )ξΩ Ωx x t x t, 0 , , , . (3.10)

One can rewrite (3.5) as a more explicit composition map type formula:

( ) [ ] ( )[ ]ξ ξ ξΩ = Ω− −t x, d , 0 d ,T 1 1 (3.11)

where ξd is the gradient of ξ, ‘T ’ represents transpose, and ‘−1’ represents inverse. In the viscous case, one can rewrite the relation (3.6) as follows

( ) [ ] ( )[ ] [ ]

[ ] ( )[ ] [ ]∫

ξ ξ ξ ξ

ξ ξ τ ξ τ ξ

Ω = Ω +

× ∆Ω

− − −

−⎜ ⎟⎛⎝

⎞⎠

t x, d , 0 d1

Red

d , d d d .

T T

tT

1 1 1

0

1

(3.12)

The first term is the same composition map type term as the inviscid case (3.11). The second term is the viscous term. The viscous term involves a temporal integration of the Laplacian of the vorticity ∆Ω. Thus, in the viscous case, ( )ξΩ t, depends on not only ( )Ω x, 0 and ( )ξ x t, ,

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but also ( ( ) )ξ τ τ∆Ω x, , , [ ]τ∈ t0, . The relation (3.12) loses the composition map type nature of the inviscid case, and it describes a map of the form

( ( ) ( ) ( [ ]) ( ( ) ) ( [ ])) ( )ξ τ τ ξ τ τ τΩ ∈ ∆Ω ∈ Ωx x t x t x t, 0 , , 0, , , , 0, , .

in contrast to (3.10) in the inviscid case. With the loss of the composition map type nature, it turns out that the solution operator of the Navier–Stokes equations is everywhere differen-tiable in contrast to the nowhere differentiability of the solution operator of the Euler equa-tions. Furthermore, the norm of the derivative of the solution operator of the Navier–Stokes equations has an upper bound characterized by viscosity (2.8), and at the explicit solutions given above, it also has a lower bound characterized by viscosity (2.16).

4. Conclusion

By constructing explicit solutions to 2D Euler and 2D Navier–Stokes equations, we showed the rough dependence of their solution operators upon initial data. The effect of viscosity is also studied. Finally we presented an extension of a theorem of Cauchy to the viscous case.

References

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